$\mathrm{x} = {\cos }^{-1}\left(\frac{1}{\sqrt{1 + {\mathrm{t}}^2}}\right), \mathrm{y} = {\sin }^{-1}\left(\frac{1}{\sqrt{1 + {\mathrm{t}}^2}}\right) \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• 0

• tan(t)

• 1

• sin(t) cos(t)

The simplified expression of sin(tan^-1(x)) for any real number x is given by

In a triangle ABC, the sides b and c are the roots of the equation x² - 61x + 820 = 0 and A = ${\tan }^{-1}\left(\frac{4}{3}\right)$, then a² is equal to

• 1098

• 1096

• 1097

• 1095

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

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\mathrm{\pi }$

• $\frac{3\mathrm{\pi }}{4}$

If ${\tan }^{-1}\left(\frac{\mathrm{m}}{\mathrm{n}}\right) - {\tan }^{-1}\left(\frac{\mathrm{m} - \mathrm{n}}{\mathrm{m} +\hspace{0.17em}\mathrm{n}}\right)$ is equal to

• ${\tan }^{-1}\left(\frac{\mathrm{n}}{\mathrm{m}}\right)$

• ${\tan }^{-1}\left(\frac{\mathrm{m} +\hspace{0.17em}\mathrm{n}}{\mathrm{m} - \mathrm{n}}\right)$

• $\frac{\mathrm{\pi }}{4}$

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

If ${\sec }^{-1}\left(\sqrt{1 + {\mathrm{x}}^2}\right) + {\csc }^{-1}\left(\frac{\sqrt{1 +\hspace{0.17em}{\mathrm{y}}^2}}{\mathrm{y}}\right) + {\cot }^{-1}\left(\frac{1}{\mathrm{z}}\right) = \mathrm{\pi }$, then x + y + z is equal to

• xyz

• 2xyz

• xyz²

• x²yz

167.

If ${\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^2 + {\left({\cot }^{-1}\left(\mathrm{x}\right)\right)}^2 = \frac{5{\mathrm{\pi }}^2}{8}$, then x is equal to

• 0

• 2

• 1

• - 1

$5{\cos }^{-1}\left(\frac{1 - {\mathrm{x}}^2}{1 + {\mathrm{x}}^2}\right) + 7{\sin }^{-1}\left(\frac{2\mathrm{x}}{1 \hspace{0.17em}+ {\mathrm{x}}^2}\right) - 4{\tan }^{-1}\left(\frac{2\mathrm{x}}{1 \hspace{0.17em}+ {\mathrm{x}}^2}\right) - {\tan }^{-1}\left(\mathrm{x}\right) = 5\mathrm{\pi }$, then x is equal to

• 3

• - $\sqrt{3}$

• $\sqrt{2}$

• $\sqrt{3}$

If ${\cos }^{-1}\left(\frac{5}{13}\right) - {\sin }^{-1}\left(\frac{12}{13}\right) = {\cos }^{-1}\left(\mathrm{x}\right)$,  then x is equal to

• 1

• $\frac{1}{\sqrt{2}}$

• 0

• $\frac{\sqrt{3}}{2}$

The value of $\cos \left[{\tan }^{-1}\left\{\sin \left({\cot }^{-1}\left(\mathrm{x}\right)\right)\right\}\right]$ is

• $\sqrt{\frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2 - 1}}$

• $\sqrt{\frac{1 - {\mathrm{x}}^2}{{\mathrm{x}}^2 + 2}}$

• $\sqrt{\frac{1 - {\mathrm{x}}^2}{1 + {\mathrm{x}}^2}}$

• $\sqrt{\frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2 + 2}}$