Question

The value of ${\cosh }^{-1}\left(\sec \left(x\right)\right)$ is

• A. $\log \left(\frac{1+\sin \left(x\right)}{\cos \left(x\right)}\right)$
• B. $\log \left(\frac{1-\sin \left(x\right)}{\cos \left(x\right)}\right)$
• C. $\log \left(\frac{1+\cos \left(x\right)}{\sin \left(x\right)}\right)$
• D. $\log \left(\frac{1-\cos \left(x\right)}{\sin \left(x\right)}\right)$

Answer 1

The correct option is A

$\log \left(\frac{1+\sin \left(x\right)}{\cos \left(x\right)}\right)$

Explanation for correct option

Given hyperbolic function is ${\cosh }^{-1}\left(\sec \left(x\right)\right)$

$$\begin{gathered} {\cosh }^{-1}\left(\sec \left(x\right)\right)=\log \left(\sec \left(x\right)+\sqrt{{\sec }^2\left(x\right)-1}\right)\left[\because {\cosh }^{-1}\left(x\right)=\log \left(x+\sqrt{x^2-1}\right)\right] \\ =\log \left(\sec \left(x\right)+\tan \left(x\right)\right)\left[\because {\sec }^2\left(x\right)-{\tan }^2\left(x\right)=1\right] \\ =\log \left(\frac{1+\sin \left(x\right)}{\cos \left(x\right)}\right) \end{gathered}$$

Hence, the correct option is $o\mathrm{ption}\left(\mathrm{A}\right)$