Inverse Trigonometric Functions - Trigonometry - Mathematics Previous Year Questions

The solution of \({\tan ^{ - 1}}x + 2{\cot ^{ - 1}}x = {{2\pi } \over 3}\) is

If \({\sec ^{ - 1}}\left( {{{1 + x} \over {1 - y}}} \right) = a\), then \({{dy} \over {dx}}\) is

The value of \(\sin \left[ {2{{\cos }^{ - 1}}{{\sqrt 5 } \over 3}} \right]\) is

$$\text { Let } f(x)=\cos ^{-1}(3 x-1) \text {, then domain of } f(x) \text { is equal to }$$

$$\text { The value of } \sin ^{-1}\left[\cos \left(39 \frac{\pi}{5}\right)\right] \text { is }$$

Evaluate :

$$\operatorname{cosec}^{-1}\left(-\frac{2 \sqrt{3}}{3}\right)+\tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right)+\sec ^{-1} 2+\cos ^{-1}\left(-\frac{1}{2}\right)-\sin ^{-1}\left(\frac{\sqrt{2}}{2}\right)$$

$$\text { If } \alpha=\tan ^{-1}\left(\tan \frac{5 \pi}{4}\right) \text { and } \beta=\tan ^{-1}\left(-\tan \frac{2 \pi}{3}\right) \text { then }$$

$$\text { If } y=\sin ^{-1}(\sqrt{\sin x}) \text {, then } \frac{d y}{d x} \text { equals }$$

$$\text { Evaluate: } \cot ^{-1}\left(-\frac{3}{\sqrt{3}}\right)-\sec ^{-1}\left(-\frac{2}{\sqrt{2}}\right)-\operatorname{cosec}^{-1}(-1)-\tan ^{-1}(1)$$

$$\text { The function } f(x)=\tan ^{-1}(\sin x+\cos x) \text { is an increasing function in }$$

$$\text { Evaluate: } \cos ^{-1}\left(\cos \frac{35 \pi}{18}\right)-\sin ^{-1}\left(\sin \frac{35 \pi}{18}\right)$$

\(\text { The value of } \sin ^{-1}\left[\cot \left(\frac{1}{2} \tan ^{-1} \frac{1}{\sqrt{3}}+\cos ^{-1} \frac{\sqrt{12}}{4}+\sin ^{-1} \frac{1}{\sqrt{2}}\right)\right]\) is

Evaluate : \(\cos ^{-1}\left[\cos \left(-680^{\circ}\right)\right]+\sin ^{-1}\left[\sin \left(-600^{\circ}\right)\right]-\cos ^{-1}\left(\sin 270^{\circ}\right)\)

Which of the following is the simplest form of the expression $\boldsymbol{\operatorname { t a n }}^{-\mathbf{1}}\left(\frac{\sqrt{\mathbf{1 + x ^ { \mathbf { 2 } }}}-\mathbf{1}}{\boldsymbol{x}}\right)$ where $x \neq 0$