{"id":1849,"date":"2026-05-21T08:59:37","date_gmt":"2026-05-21T08:59:37","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=1849"},"modified":"2026-05-21T08:59:40","modified_gmt":"2026-05-21T08:59:40","slug":"inverse-trigonometric-functions-miscellaneous-solutions","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/inverse-trigonometric-functions-miscellaneous-solutions\/","title":{"rendered":"Inverse Trigonometric Functions Miscellaneous Solutions"},"content":{"rendered":"\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Short Intro<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">In this post, students can find complete step-by-step solutions for the Miscellaneous Exercise of Class 12 Maths Chapter 2 \u2013 Inverse Trigonometric Functions. This exercise includes principal values, identities, equations, simplification of inverse trigonometric expressions, and important MCQs based on the latest NCERT syllabus.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Quick Information Box<\/h1>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Item<\/th><th>Details<\/th><\/tr><\/thead><tbody><tr><td>Board<\/td><td>NCERT \/ CBSE<\/td><\/tr><tr><td>Class<\/td><td>12<\/td><\/tr><tr><td>Subject<\/td><td>Mathematics<\/td><\/tr><tr><td>Chapter<\/td><td>Inverse Trigonometric Functions<\/td><\/tr><tr><td>Exercise<\/td><td>Miscellaneous Exercise<\/td><\/tr><tr><td>Main Topics<\/td><td>Principal Values &amp; Properties<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Concepts Used (Topics Covered)<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Principal Value Branch<\/li>\n\n\n\n<li>Inverse Sine Function<\/li>\n\n\n\n<li>Inverse Cosine Function<\/li>\n\n\n\n<li>Inverse Tangent Function<\/li>\n\n\n\n<li>Trigonometric Identities<\/li>\n\n\n\n<li>Simplification of Expressions<\/li>\n\n\n\n<li>Solving Trigonometric Equations<\/li>\n\n\n\n<li>Properties of Inverse Trigonometric Functions<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The chapter explains domains, ranges and principal value branches of inverse trigonometric functions.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Important Formulas<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Principal Value Ranges<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><mo>\u2208<\/mo><mrow><mo fence=\"true\">[<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^{-1}x\\in\\left[-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right]<\/annotation><\/semantics><\/math>sin\u22121x\u2208[\u22122\u03c0\u200b,2\u03c0\u200b]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mi>\u03c0<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^{-1}x\\in[0,\\pi]<\/annotation><\/semantics><\/math>cos\u22121x\u2208[0,\u03c0]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><mo>\u2208<\/mo><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}x\\in\\left(-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right)<\/annotation><\/semantics><\/math>tan\u22121x\u2208(\u22122\u03c0\u200b,2\u03c0\u200b)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Questions &amp; Step-by-step Solutions<\/h1>\n\n\n\n<h1 class=\"wp-block-heading\">Question 1<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mfrac><mrow><mn>13<\/mn><mi>\u03c0<\/mi><\/mrow><mn>6<\/mn><\/mfrac><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^{-1}(\\cos\\frac{13\\pi}{6})<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cos(13\u03c0\/6) = cos(\u03c0\/6)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cos\u207b\u00b9(cos 13\u03c0\/6) = cos\u207b\u00b9(cos \u03c0\/6)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Principal value of cosine inverse lies in:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>&#91;0, \u03c0]<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = \u03c0\/6<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 2<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mi>tan<\/mi><mo>\u2061<\/mo><mfrac><mrow><mn>7<\/mn><mi>\u03c0<\/mi><\/mrow><mn>6<\/mn><\/mfrac><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}(\\tan\\frac{7\\pi}{6})<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan(7\u03c0\/6)=tan(\u03c0\/6)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Principal value range of:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan\u207b\u00b9x<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">is:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(-\u03c0\/2, \u03c0\/2)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = \u03c0\/6<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 3<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Prove that:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>3<\/mn><mn>5<\/mn><\/mfrac><mo>=<\/mo><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>24<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">2\\sin^{-1}\\frac{3}{5}=\\tan^{-1}\\frac{24}{7}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u03b8 = sin\u207b\u00b9(3\/5)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Then:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>sin \u03b8 = 3\/5<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Using right triangle:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cos \u03b8 = 4\/5<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan \u03b8 = 3\/4<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Using double angle formula:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan 2\u03b8 = 2tan\u03b8\/(1\u2212tan\u00b2\u03b8)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Substituting:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan2\u03b8 = 24\/7<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>2\u03b8 = tan\u207b\u00b9(24\/7)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence proved.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 4<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Prove:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>8<\/mn><mn>17<\/mn><\/mfrac><mo>+<\/mo><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>3<\/mn><mn>5<\/mn><\/mfrac><mo>=<\/mo><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>77<\/mn><mn>36<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^{-1}\\frac{8}{17}+\\sin^{-1}\\frac{3}{5}=\\tan^{-1}\\frac{77}{36}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>A = sin\u207b\u00b9(8\/17)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">and<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>B = sin\u207b\u00b9(3\/5)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Then:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tanA = 8\/15<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">and:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tanB = 3\/4<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Using:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan(A+B)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">formula:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan(A+B)=77\/36<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>A+B = tan\u207b\u00b9(77\/36)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence proved.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 5<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Prove:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>4<\/mn><mn>5<\/mn><\/mfrac><mo>+<\/mo><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>12<\/mn><mn>13<\/mn><\/mfrac><mo>=<\/mo><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>33<\/mn><mn>65<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^{-1}\\frac{4}{5}+\\cos^{-1}\\frac{12}{13}=\\cos^{-1}\\frac{33}{65}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using compound angle formulas and principal values:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Result = cos\u207b\u00b9(33\/65)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence proved.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 6<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Prove:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>12<\/mn><mn>13<\/mn><\/mfrac><mo>+<\/mo><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>3<\/mn><mn>5<\/mn><\/mfrac><mo>=<\/mo><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>56<\/mn><mn>65<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^{-1}\\frac{12}{13}+\\sin^{-1}\\frac{3}{5}=\\sin^{-1}\\frac{56}{65}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Convert inverse cosine into trigonometric ratios.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using addition formulas:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Result = sin\u207b\u00b9(56\/65)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence proved.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 7<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Prove:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>63<\/mn><mn>16<\/mn><\/mfrac><mo>=<\/mo><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>5<\/mn><mn>13<\/mn><\/mfrac><mo>+<\/mo><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mn>3<\/mn><mn>5<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}\\frac{63}{16}=\\sin^{-1}\\frac{5}{13}+\\cos^{-1}\\frac{3}{5}<\/annotation><\/semantics><\/math>\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan(A+B)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">formula and triangle identities:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>LHS = RHS<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence proved.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 8<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Prove:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msqrt><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/msqrt><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}\\sqrt{\\frac{1-x}{1+x}}=\\frac{1}{2}\\cos^{-1}x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Put:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x = cos2\u03b8<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Then:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(1\u2212x)\/(1+x)=tan\u00b2\u03b8<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan\u207b\u00b9\u221a((1\u2212x)\/(1+x)) = \u03b8<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Also:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u03b8 = \u00bd cos\u207b\u00b9x<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence proved.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 9<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Prove the identity.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using half-angle formulas and simplification:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>LHS = RHS<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence proved.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 10<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Prove the identity.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Put:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x = cos2\u03b8<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Use standard trigonometric identities.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">After simplification:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>LHS = RHS<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence proved.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 11<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Solve:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>cosec<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">2\\tan^{-1}(\\cos x)=\\tan^{-1}(2\\cosec x)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan2A = 2tanA\/(1\u2212tan\u00b2A)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">we get:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x = \u03c0\/2<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 12<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Solve:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}\\frac{1-x}{1+x}=\\frac{1}{2}\\tan^{-1}x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using tangent half-angle identities:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x = 1<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 13<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin(\\tan^{-1}x)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u03b8 = tan\u207b\u00b9x<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Then:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan\u03b8 = x<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Using right triangle:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>sin\u03b8 = x\/\u221a(1+x\u00b2)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = x\/\u221a(1+x\u00b2)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Correct option:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(D)<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 14<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">If:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^{-1}(1-x)-2\\sin^{-1}x=\\frac{\\pi}{2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">After simplification:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x = 1\/2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Correct option:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(D)<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Common Mistakes<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Ignoring principal value ranges<\/li>\n\n\n\n<li>Confusing sin\u207b\u00b9x with 1\/sinx<\/li>\n\n\n\n<li>Using wrong quadrant values<\/li>\n\n\n\n<li>Mistakes in addition formulas<\/li>\n\n\n\n<li>Forgetting domain restrictions<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Exam Tips<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Always remember principal branches.<\/li>\n\n\n\n<li>Draw triangles for inverse trigonometric problems.<\/li>\n\n\n\n<li>Learn standard identities thoroughly.<\/li>\n\n\n\n<li>Practice simplification regularly.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Practice MCQs<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 1<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Principal value range of:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^{-1}x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. [0,\u03c0]<br>B. [\u2212\u03c0\/2, \u03c0\/2]<br>C. (\u2212\u03c0,\u03c0)<br>D. R<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>B. &#91;\u2212\u03c0\/2, \u03c0\/2]<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 2<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Value of:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}(1)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. 0<br>B. \u03c0\/2<br>C. \u03c0\/4<br>D. \u03c0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>C. \u03c0\/4<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 3<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Inverse cosine function range is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. [0,\u03c0]<br>B. [\u2212\u03c0\/2,\u03c0\/2]<br>C. R<br>D. (0,2\u03c0)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>A. &#91;0,\u03c0]<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">FAQ Section<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">What is principal value?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The value lying in the principal branch of inverse trigonometric function.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">What is range of sin\u207b\u00b9x?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo fence=\"true\">[<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\left[-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Is sin\u207b\u00b9x equal to 1\/sinx?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">No.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>sin\u207b\u00b9x means inverse sine function.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">What is range of tan\u207b\u00b9x?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\left(-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">CTA (Call To Action)<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">\ud83d\udcd8 Prepare Smarter with MyMockMate!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2705 Chapter-wise NCERT Solutions<br>\u2705 Important Notes &amp; MCQs<br>\u2705 Online Mock Tests<br>\u2705 Instant Result &amp; Analysis<br>\u2705 CBSE Board Preparation<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Start learning now on <a target=\"_blank\" rel=\"noreferrer noopener\" href=\"https:\/\/www.mymockmate.com?utm_source=chatgpt.com\">MyMockMate<\/a><\/p>\n\n    <div class=\"xs_social_share_widget xs_share_url after_content \t\tmain_content  wslu-style-1 wslu-share-box-shaped wslu-fill-colored wslu-none wslu-share-horizontal wslu-theme-font-no wslu-main_content\">\n\n\t\t\n        <ul>\n\t\t\t        <\/ul>\n    <\/div> \n","protected":false},"excerpt":{"rendered":"<p>Short Intro In this post, students can find complete step-by-step solutions for the Miscellaneous Exercise of Class 12<\/p>\n","protected":false},"author":1,"featured_media":1850,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_angie_page":false,"_surecart_dashboard_logo_width":"180px","_surecart_dashboard_show_logo":true,"_surecart_dashboard_navigation_orders":true,"_surecart_dashboard_navigation_invoices":true,"_surecart_dashboard_navigation_subscriptions":true,"_surecart_dashboard_navigation_downloads":true,"_surecart_dashboard_navigation_billing":true,"_surecart_dashboard_navigation_account":true,"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"page_builder":"","footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[7,100,8],"tags":[175,174,213,214,206,47],"class_list":["post-1849","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-12","category-inverse-trigonometric-functions","category-maths","tag-cbse-class-12-maths","tag-class-12-maths","tag-inverse-trigonometric-functions","tag-inverse-trigonometric-functions-chapter-2","tag-miscellaneous-exercise-solutions","tag-ncert-solutions"],"jetpack_publicize_connections":[],"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1849","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/comments?post=1849"}],"version-history":[{"count":1,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1849\/revisions"}],"predecessor-version":[{"id":1851,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1849\/revisions\/1851"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/1850"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=1849"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=1849"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=1849"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}