Mathematics : Inverse Trigonometric Functions

${\sinh }^{-1}\left(2\right) + {\sinh }^{-1}\left(3\right) = \mathrm{x} \Rightarrow \cosh \left(\mathrm{x}\right) = ?$

• $\frac{1}{2}\left(3\sqrt{5} + 2\sqrt{10}\right)$

• $\frac{1}{2}\left(3\sqrt{5} - 2\sqrt{10}\right)$

• $\frac{1}{2}\left(12 + 2\sqrt{50}\right)$

• $\frac{1}{2}\left(12 - 2\sqrt{50}\right)$

$\mathrm{In} \mathrm{any} ∆\mathrm{ABC}, \mathrm{a}\left(\mathrm{bcos}\left(\mathrm{C}\right) - \mathrm{ccos}\left(\mathrm{B}\right)\right) = ?$

• b² + c²

• b² - c²

• $\frac{1}{\mathrm{b}} + \frac{1}{\mathrm{c}}$

• $\frac{1}{{\mathrm{b}}^2} - \frac{1}{{\mathrm{c}}^2}$

$\mathrm{If} {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) + {\tan }^{-1}\left(\mathrm{z}\right) = \frac{\mathrm{\pi }}{2}, \mathrm{then} 1 - \mathrm{xy} - \mathrm{yz} - \mathrm{zx} = ?$

• 1

• 0

• - 1

• 2

$\mathrm{If} {\tanh }^{-1}\left(\mathrm{x}\right) = \mathrm{alog}\left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right), \left|\mathrm{x}\right| < 1, \mathrm{then} \mathrm{a} = ?$

• 1

• 2

• $\frac{1}{2}$

• $\frac{1}{4}$

${\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^2 + {\left({\cot }^{-1}\left(\mathrm{x}\right)\right)}^2 = \frac{5{\mathrm{\pi }}^2}{8} \Rightarrow \mathrm{x} =$

• - 1

• 1

• 0

• $\mathrm{\pi }\sqrt{\frac{5}{8}}$

$\mathrm{For} 0 < \mathrm{x} \le \mathrm{\pi }, {\sinh }^{-1}\left(\cot \left(\mathrm{x}\right)\right) = ?$

• $\log \left(\cot \left(\frac{\mathrm{x}}{2}\right)\right)$

• $\log \left(\tan \left(\frac{\mathrm{x}}{2}\right)\right)$

• $\log \left(1 + \cot \left(\mathrm{x}\right)\right)$

• $\log \left(1 + \tan \left(\mathrm{x}\right)\right)$

$$\begin{gathered} \mathrm{If} \frac{1}{2} \le \mathrm{x} \le 1, \mathrm{then} \\ {\cos }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\frac{\mathrm{x}}{2} + \frac{1}{2}\sqrt{3 - 3{\mathrm{x}}^2}\right) = ? \end{gathered}$$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\mathrm{\pi }$

• 0

$\mathrm{If} {\cos }^{-1}\left(\frac{5}{13}\right) + {\cos }^{-1}\left(\frac{3}{5}\right) = {\cos }^{-1}\left(\mathrm{x}\right), \mathrm{then} \mathrm{x} = ?$

• $\frac{3}{63}$

• $\frac{- 36}{65}$

• $\frac{- 33}{65}$

• - 1

${\tanh }^{-1}\left(\frac{1}{2}\right) + {\coth }^{-1}\left(2\right) = ?$

• $\frac{1}{3}\log \left(3\right)$

• $\frac{1}{2}\log \left(6\right)$

• $\frac{1}{2}\log \left(12\right)$

• $\log \left(3\right)$

$$\begin{gathered} \mathrm{If} \mathrm{x} > 0, \mathrm{y} > 0, \mathrm{z} > 0, \mathrm{xy} + \mathrm{yz} + \mathrm{zx} < 1 \mathrm{and} \mathrm{if} {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) + {\tan }^{-1}\left(\mathrm{z}\right) = \mathrm{\pi }, \\ \mathrm{then} \mathrm{x} + \mathrm{y} +\hspace{0.17em}\mathrm{z} =? \end{gathered}$$

• 0

• xyz

• 3xyz

• $\sqrt{\mathrm{xyz}}$