Question

Evaluate: $sec{h}^{-1}\left(\frac{1}{2}\right)$

• A. $\log \left(\sqrt{3}+\sqrt{2}\right)$
• B. $\log \left(\sqrt{3}+1\right)$
• C. $\log \left(2+\sqrt{3}\right)$
• D. None of these

Answer 1

The correct option is C

$\log \left(2+\sqrt{3}\right)$

Explanation for the correct option:

Using the hyperbolic cosine relationship for the given expression:

$\begin{array}{rcl}\sec h^{-1}\left(\frac{1}{2}\right)& =& \cos h^{-1}\left(2\right)\left[\because \sec h^{-1}\left(\frac{1}{x}\right)=\cos h^{-1}\left(x\right)\right]\\ & =& \log \left(2+\sqrt{\left.2^2-1\right)}\right.\left(\because {\cosh }^{-1}x={\log }_n\left(x+\sqrt{x^2-1}\right)\right)\\ & =& \log (2+\sqrt{3})\end{array}$

Hence, option (C) is correct .