NCERT Exemplar Solutions
Class 12 - Mathematics

Chapter 2: INVERSE TRIGONOMETRIC FUNCTIONS

Find the value of \( \tan^{-1}(\tan 5\pi/6) + \cos^{-1}(\cos 13\pi/6) \).

Evaluate \( \cos[\cos^{-1}(-\sqrt{3}/2) + \pi/6] \).

Prove that \( \cot(\pi/4 - 2\cot^{-1}3) = 7 \).

Find the value of \( \tan^{-1}(-1/\sqrt{3}) + \cot^{-1}(1/\sqrt{3}) + \tan^{-1}(\sin(-\pi/2)) \).

Find the value of \( \tan^{-1}(\tan 2\pi/3) \).

Show that \( 2\tan^{-1}(-3) = -\pi/2 + \tan^{-1}(-4/3) \).

Find the real solutions of the equation \( \tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^{2}+x+1} = \pi/2 \).

Find the value of the expression \( \sin(2\tan^{-1}(1/3)) + \cos(\tan^{-1}(2\sqrt{2})) \).

If \( 2\tan^{-1}(\cos \theta) = \tan^{-1}(2\csc \theta) \), then show that \( \theta = \pi/4 \).

Show that \( \cos(2\tan^{-1}(1/7)) = \sin(4\tan^{-1}(1/3)) \).

Solve the equation \( \cos(\tan^{-1} x) = \sin(\cot^{-1}(3/4)) \).

Prove that \(\tan^{-1}\left(\dfrac{\sqrt{1+x^{2}} + \sqrt{1-x^{2}}}{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}\right) = \dfrac{\pi}{4} + \dfrac{1}{2} \cos^{-1} x^{2}\).

Find the simplified form of \(\cos^{-1}\left(\dfrac{3}{5}\cos x + \dfrac{4}{5}\sin x\right)\), where \(x \in \left[ -\dfrac{3\pi}{4}, \dfrac{\pi}{4} \right]\).

Prove that \(\sin^{-1}\dfrac{8}{17} + \sin^{-1}\dfrac{3}{5} = \sin^{-1}\dfrac{77}{85}\).

Show that \(\sin^{-1}\dfrac{5}{13} + \cos^{-1}\dfrac{3}{5} = \tan^{-1}\dfrac{63}{16}\).

Prove that \(\tan^{-1}\dfrac{1}{4} + \tan^{-1}\dfrac{2}{9} = \sin^{-1}\dfrac{1}{\sqrt{5}}\).

Find the value of \(4 \tan^{-1}\dfrac{1}{5} - \tan^{-1}\dfrac{1}{239}\).

Show that \(\tan\left(\dfrac{1}{2}\sin^{-1}\dfrac{3}{4}\right) = \dfrac{4 - \sqrt{7}}{3}\) and justify why the other value \(\dfrac{4 + \sqrt{7}}{3}\) is ignored.

If \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\) is an arithmetic progression with common difference \(d\), evaluate the expression:

\[ \tan \left[ \tan^{-1}\left(\dfrac{d}{1 + a_{1}a_{2}}\right) + \tan^{-1}\left(\dfrac{d}{1 + a_{2}a_{3}}\right) + \tan^{-1}\left(\dfrac{d}{1 + a_{3}a_{4}}\right) + \cdots + \tan^{-1}\left(\dfrac{d}{1 + a_{n-1}a_{n}}\right) \right] \]

The principal value of \(\cos^{-1}(-\dfrac{1}{2})\) is ____.

The value of \(\sin^{-1}\left(\sin \dfrac{3\pi}{5}\right)\) is ____.

If \(\cos(\tan^{-1} x + \cot^{-1} \sqrt{3}) = 0\), then value of \(x\) is ____.

The set of values of \(\sec^{-1}\left(\dfrac{1}{2}\right)\) is ____.

The principal value of \(\tan^{-1} \sqrt{3}\) is ____.

The value of \(\cos^{-1}(\cos \dfrac{14\pi}{3})\) is ____.

The value of \(\cos(\sin^{-1} x + \cos^{-1} x)\), \(|x| \le 1\), is ____.

The value of the expression \(\tan\left(\dfrac{\sin^{-1} x + \cos^{-1} x}{2}\right)\), when \(x = \dfrac{\sqrt{3}}{2}\), is ____.

If \(y = 2 \tan^{-1} x + \sin^{-1}\left(\dfrac{2x}{1 + x^{2}}\right)\) for all \(x\), then ____ < y < ____.

The result \(\tan^{-1} x - \tan^{-1} y = \tan^{-1}\left(\dfrac{x - y}{1 + xy}\right)\) is true when value of \(xy\) is ____.

The value of \(\cot^{-1}(-x)\) for all \(x \in \mathbb{R}\) in terms of \(\cot^{-1} x\) is ____.

All trigonometric functions have inverse over their respective domains.

The value of the expression \((\cos^{-1} x)^2\) is equal to \(\sec^{2} x\).

The domain of trigonometric functions can be restricted to any one of their branches (not necessarily principal value) in order to obtain their inverse functions.

The least numerical value, either positive or negative, of angle \(\theta\) is called the principal value of the inverse trigonometric function.

The graph of an inverse trigonometric function can be obtained from the graph of its corresponding trigonometric function by interchanging \(x\) and \(y\) axes.

The minimum value of \(n\) for which \(\tan^{-1} \dfrac{n}{\pi} = \dfrac{\pi}{4}\), \(n \in \mathbb{N}\), is valid is 5.

The principal value of \(\sin^{-1}\left[\cos\left(\sin^{-1} \dfrac{1}{2}\right)\right]\) is \(\dfrac{\pi}{3}\).