Mathematics : Inverse Trigonometric Functions

If $\mathrm{a} \le {\sin }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{x}\right) \le \mathrm{b}$, then

• $\mathrm{a} = 0, \mathrm{b} = \mathrm{\pi }$

• $\mathrm{a} = 0, \mathrm{b} = \frac{\mathrm{\pi }}{2}$

• $\mathrm{a} = \frac{\mathrm{\pi }}{2}, \mathrm{b} = \mathrm{\pi }$

• None of these

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(\mathrm{y}\right) = \frac{2\mathrm{\pi }}{3}$ and ${\cos }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{y}\right) = \frac{\mathrm{\pi }}{3}$. Then, (x, y) is equal to

• (0, 1)

• (1/2, 1)

• (1, 1/2)

• $\left(\sqrt{3}/2, 1\right)$

If $\cot \left({\cos }^{-1}\left(\mathrm{x}\right)\right) = \sec \left({\tan }^{-1}\left(\frac{\mathrm{a}}{\sqrt{{\mathrm{b}}^2 - {\mathrm{a}}^2}}\right)\right)$, then x is equal to

• $\frac{\mathrm{b}}{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}$

• $\frac{\mathrm{a}}{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}$

• $\frac{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}{\mathrm{a}}$

• $\frac{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}{\mathrm{b}}$

Number of solutions ofthe equation

${\tan }^{-1}\left(\frac{1}{2\mathrm{x} + 1}\right) + {\tan }^{-1}\left(\frac{1}{4\mathrm{x} + 1}\right) = {\tan }^{-1}\left(\frac{2}{{\mathrm{x}}^2}\right)$ is

• 1

• 2

• 3

• 4

The value of ${\tan }^{-1}\left(\frac{1}{2}\right) + {\tan }^{-1}\left(\frac{1}{3}\right) + {\tan }^{-1}\left(\frac{7}{8}\right)$ is

• ${\tan }^{-1}\left(\frac{7}{8}\right)$

• ${\cot }^{-1}\left(15\right)$

• ${\tan }^{-1}\left(15\right)$

• ${\tan }^{-1}\left(\frac{25}{24}\right)$

If ${\tan }^{-1}\left(\mathrm{x} - 1\right) + {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{x} + 1\right) = {\tan }^{-1}\left(3\mathrm{x}\right)$, then x is

• $\pm \frac{1}{2}$

• $0, \frac{1}{2}$

• $0, - \frac{1}{2}$

• $0, \pm \frac{1}{2}$

The domain of the function ${\sin }^{-1}\left({\log }_2\left(\frac{{\mathrm{x}}^2}{2}\right)\right)$ is

• [- 1, 2] - {0}

• [- 2, 2] - (- 1, 1)

• [- 2, 2] - {0}

• [1, 2]

$3{\tan }^{-1}\left(\mathrm{a}\right)$ is equal to

• ${\tan }^{-1}\left(\frac{3\mathrm{a} + {\mathrm{a}}^3}{1 + 3{\mathrm{a}}^2}\right)$

• ${\tan }^{-1}\left(\frac{3\mathrm{a} - {\mathrm{a}}^3}{1 + 3{\mathrm{a}}^2}\right)$

• ${\tan }^{-1}\left(\frac{3\mathrm{a} + {\mathrm{a}}^3}{1 - 3{\mathrm{a}}^2}\right)$

• ${\tan }^{-1}\left(\frac{3\mathrm{a} - {\mathrm{a}}^3}{1 - 3{\mathrm{a}}^2}\right)$

The value of $\sin \left({\sin }^{-1}\left(\frac{1}{3}\right) + {\sec }^{-1}\left(3\right)\right) + \cos \left({\tan }^{-1}\left(\frac{1}{2}\right) + {\tan }^{-1}\left(2\right)\right)$ is :

• 1

• 2

• 3

• 4

The value of ${\cot }^{-1}\left(9\right) + {\csc }^{-1}\left(\frac{\sqrt{41}}{4}\right)$ is given by :

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• ${\tan }^{-1}\left(2\right)$