Inverse trigonometric functions and their properties — practice questions

Question 1 · difficulty L2 · understanding

If $\sin \left(\tan ^{-1}(x \sqrt{2})\right)=\cot \left(\sin ^{-1} \sqrt{1-x^2}\right), x \in(0,1)$, then the value of $x$ is:

• A. $\frac{1}{2}$
• B. ${\frac{1}{3}}$
• C. $\frac{2}{3}$
• D. $\frac{5}{8}$

Question 2 · difficulty L2 · understanding

The value of $${\tan ^{ - 1}}\left( {{{\cos \left( {{{15\pi } \over 4}} \right) - 1} \over {\sin \left( {{\pi \over 4}} \right)}}} \right)$$ is equal to :

• A. $$- {\pi \over 4}$$
• B. $$- {\pi \over 8}$$
• C. $$- {{5\pi } \over {12}}$$
• D. $$- {{4\pi } \over 9}$$

Question 3 · difficulty L2 · understanding

$${\sin ^1}\left( {\sin {{2\pi } \over 3}} \right) + {\cos ^{ - 1}}\left( {\cos {{7\pi } \over 6}} \right) + {\tan ^{ - 1}}\left( {\tan {{3\pi } \over 4}} \right)$$ is equal to :

• A. $${{11\pi } \over {12}}$$
• B. $${{17\pi } \over {12}}$$
• C. $${{31\pi } \over {12}}$$
• D. $$-$$$${{3\pi } \over {4}}$$

Question 4 · difficulty L2 · understanding

Considering only the principal values of the inverse trigonometric functions, the value of $$\cot^{-1}(\cot(-11)) + 10 \sin\left(2 \cos^{-1}\left(\frac{1}{\sqrt{2}}\right)\right) + 10\sin(2 \tan^{-1}(2))$$ is

• A. $3\pi + 7$
• B. $7$
• C. $4\pi + 7$
• D. $3\pi - 5$

Question 5 · difficulty L3 · understanding

If $k=\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{2}{3}\right)\right)+\tan \left(\frac{1}{2} \sin ^{-1}\left(\frac{2}{3}\right)\right)$, then the number of solutions of the equation $\sin ^{-1}(k x-1)=\sin ^{-1} x-\cos ^{-1} x$ is $\_\_\_\_$.

Question 6 · difficulty L3 · understanding

Let the maximum value of $\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2$ for $x \in\left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\right]$ be $\frac{\mathrm{m}}{\mathrm{n}} \pi^2$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to $\_\_\_\_$。

Question 7 · difficulty L3 · understanding

$$\text { If } y=\cos \left(\frac{\pi}{3}+\cos ^{-1} \frac{x}{2}\right) \text {, then }(x-y)^2+3 y^2 \text { is equal to }$$

Question 8 · difficulty L3 · understanding

Let S = $\left\{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1} [2x + 1] \right\}$. Then $\sum\limits_{x \in S} (2x - 1)^2$ is equal to _______.

Question 9 · difficulty L3 · understanding

If for some $\alpha, \beta ; \alpha \leq \beta, \alpha+\beta=8$ and $\sec ^2\left(\tan ^{-1} \alpha\right)+\operatorname{cosec}^2\left(\cot ^{-1} \beta\right)=36$, then $\alpha^2+\beta$ is __________

Question 10 · difficulty L3 · understanding

For $$n \in \mathrm{N}$$, if $$\cot ^{-1} 3+\cot ^{-1} 4+\cot ^{-1} 5+\cot ^{-1} n=\frac{\pi}{4}$$, then $$n$$ is equal to ________.