# Exercise 2.2 Inverse Trigonometric Functions Class 12 Maths

## Topics/Questions discussed in this video:

00:00:00 Introduction for inverse trigonometry identities

00:03:34 Derivation of all inverse trigonometric identities

Prove the following:

00:15:14 NCERT Solutions Exercise 2.2 Question 1 3{\sin ^{ - 1}}x = {\sin ^{ - 1}}(3x - 4{x^3}),\,\,x \in \left[ { - \frac{1}{2},\,\,\frac{1}{2}} \right]

00:20:04 NCERT Solutions Exercise 2.2 Question 2 3{\cos ^{ - 1}}x = {\cos ^{ - 1}}(3x - 4{x^3}),\,\,x \in \left[ {\frac{1}{2},\,\,1} \right]

00:21:34 NCERT Solutions Exercise 2.2 Question 3 {\tan ^{ - 1}}\frac{2}{{11}} + {\tan ^{ - 1}}\frac{7}{{24}} = {\tan ^{ - 1}}\frac{1}{2}

00:23:44 NCERT Solutions Exercise 2.2 Question 4 2{\tan ^{ - 1}}\frac{1}{2} + {\tan ^{ - 1}}\frac{1}{7} = {\tan ^{ - 1}}\frac{{31}}{{17}}

Write the following functions in the simplest form:

00:27:54 NCERT Solutions Exercise 2.2 Question 6 {\tan ^{ - 1}}\left( {\frac{1}{{\sqrt {{x^2} - 1} }}} \right),\,\,|x| > 1

00:34:54 NCERT Solutions Exercise 2.2 Question 7 {\tan ^{ - 1}}\left( {\sqrt {\frac{{1 - \cos x}}{{1 + \cos x}}} } \right),\,\,0 < x < \pi

00:36:24 NCERT Solutions Exercise 2.2 Question 8 {\tan ^{ - 1}}\left( {\frac{{\cos x - \sin x}}{{\cos x + \sin x}}} \right),\,\,\frac{{ - \pi }}{4} < x < \frac{{3\pi }}{4}

00:38:54 NCERT Solutions Exercise 2.2 Question 5 {\tan ^{ - 1}}\left( {\frac{{\sqrt {1 + {x^2}}  - 1}}{x}} \right),\,\,x \ne 0

00:42:54 NCERT Solutions Exercise 2.2 Question 9 {\tan ^{ - 1}}\left( {\frac{x}{{\sqrt {{a^2} - {x^2}} }}} \right),\,\,|x|\,\, < \,a

00:45:44 NCERT Solutions Exercise 2.2 Question 10 {\tan ^{ - 1}}\left( {\frac{{3{a^2}x - {x^3}}}{{{a^3} - 3a{x^2}}}} \right),\,\,a > 0;\,\,\frac{{ - a}}{{\sqrt 3 }} < x < \frac{a}{{\sqrt 3 }}

Find the values of each of the following:

00:48:54 NCERT Solutions Exercise 2.2 Question 11 {\tan ^{ - 1}}\left[ {2\cos \left( {2{{\sin }^{ - 1}}\frac{1}{2}} \right)} \right]

00:50:24 NCERT Solutions Exercise 2.2 Question 12 \cot ({\tan ^{ - 1}}a + {\cot ^{ - 1}}a)

00:51:04 NCERT Solutions Exercise 2.2 Question 13 \tan \frac{1}{2}\left[ {{{\sin }^{ - 1}}\frac{{2x}}{{1 + {x^2}}} + {{\cos }^{ - 1}}\frac{{1 - {y^2}}}{{1 + {y^2}}}} \right],\,\,|x| < 1,\,\,y > 0\,\,{\rm{and}}\,\,xy < 1

00:52:54 NCERT Solutions Exercise 2.2 Question 14 If \sin \left( {{{\sin }^{ - 1}}\frac{1}{5} + {{\cos }^{ - 1}}x} \right) = 1, then find the value of x.

00:54:44 NCERT Solutions Exercise 2.2 Question 15 If {\tan ^{ - 1}}\frac{{x - 1}}{{x - 2}} + {\tan ^{ - 1}}\frac{{x + 1}}{{x + 2}} = \frac{\pi }{4} then find the value of x.

Find the values of each of the expressions in Exercises 16 to 18.

00:59:54 NCERT Solutions Exercise 2.2 Question 16 {\sin ^{ - 1}}\left( {\sin \frac{{2\pi }}{3}} \right)

01:01:24 NCERT Solutions Exercise 2.2 Question 17 {\tan ^{ - 1}}\left( {\tan \frac{{3\pi }}{4}} \right)

01:03:04 NCERT Solutions Exercise 2.2 Question 19 {\cos ^{ - 1}}\left( {\cos \frac{{7\pi }}{6}} \right)

(A) \frac{{7\pi }}{6}
(B) \frac{{5\pi }}{6}
(C) \frac{\pi }{3}
(D) \frac{\pi }{6}

01:04:24 NCERT Solutions Exercise 2.2 Question 20 \sin \left( {\frac{\pi }{3} - {{\sin }^{ - 1}}\left( { - \frac{1}{2}} \right)} \right)

(A) \frac{1}{2}
(B) \frac{1}{3}
(C) \frac{1}{4}
(D) 1

01:05:34 NCERT Solutions Exercise 2.2 Question 21 {\tan ^{ - 1}}\sqrt 3  - {\cot ^{ - 1}}\left( { - \sqrt 3 } \right)

(A) \pi
(B) - \frac{\pi }{2}
(C) 0
(D) 2\sqrt 3

01:07:04 NCERT Solutions Exercise 2.2 Question 18 \tan \left( {{{\sin }^{ - 1}}\frac{3}{5} + {{\cot }^{ - 1}}\frac{3}{2}} \right)

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