NCERT Solutions
Class 12 - Mathematics Part-1 - Chapter 2: INVERSE TRIGONOMETRIC FUNCTIONS

Miscellaneous Exercise on Chapter 2

Find the value of the following:

\( \cos^{-1}(\cos \frac{13\pi}{6}) \)

Find the value of the following:

\( \tan^{-1}(\tan \frac{7\pi}{6}) \)

Prove that:

\( 2\sin^{-1}\frac{3}{5} = \tan^{-1}\frac{24}{7} \)

Prove that:

\( \sin^{-1}\frac{8}{17} + \sin^{-1}\frac{3}{5} = \tan^{-1}\frac{77}{36} \)

Prove that:

\( \cos^{-1}\frac{4}{5} + \cos^{-1}\frac{12}{13} = \cos^{-1}\frac{33}{65} \)

Prove that:

\( \cos^{-1}\frac{12}{13} + \sin^{-1}\frac{3}{5} = \sin^{-1}\frac{56}{65} \)

Prove that:

\( \tan^{-1}\frac{63}{16} = \sin^{-1}\frac{5}{13} + \cos^{-1}\frac{3}{5} \)

Prove that:

\( \tan^{-1}\sqrt{x} = \frac{1}{2}\cos^{-1}\frac{1-x}{1+x}, \; x \in [0,1] \)

Prove that:

\( \cot^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right) = \frac{x}{2}, \; x \in \left(0,\frac{\pi}{4}\right) \)

Prove that:

\( \tan^{-1}\left(\sqrt{\frac{1+x}{1-x}}\right) = \frac{\pi}{4} + \frac{1}{2}\cos^{-1}x, \; -\frac{1}{\sqrt{2}} \le x \le 1 \)

Solve the equation:

\( 2\tan^{-1}(\cos x) = \tan^{-1}(2\csc x) \)

Solve the equation:

\( \tan^{-1}\frac{1-x}{1+x} = \frac{1}{2}\tan^{-1}x, \; x > 0 \)

\( \sin(\tan^{-1}x), |x| < 1 \) is equal to:

(A) \( \frac{x}{\sqrt{1-x^2}} \)

(B) \( \frac{1}{\sqrt{1-x^2}} \)

(C) \( \frac{1}{\sqrt{1+x^2}} \)

(D) \( \frac{x}{\sqrt{1+x^2}} \)

If \( \sin^{-1}(1-x) - 2\sin^{-1}x = \frac{\pi}{2} \), then \( x \) is equal to:

(A) \(0, \frac{1}{2}\)

(B) \(1, \frac{1}{2}\)

(C) 0

(D) \( \frac{1}{2} \)