Question

$arcsech\left(\frac{1}{2}\right)$ is equal to

• A. $\log (\surd 3+\surd 2)$
• B. $\log (\surd 3+1)$
• C. $\log (2+\surd 3)$
• D. None of these

Answer 1

The correct option is C

$\log (2+\surd 3)$

Explanation for the correct option:

As we know,

$\mathit{a}\mathit{r}\mathit{c}\mathit{s}\mathit{e}\mathit{c}\mathbf{}\mathit{h}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathit{l}\mathit{o}\mathit{g}\left[\frac{1+\sqrt{(1–x^2)}}{x}\right]$

Substituting $x=\frac{1}{2}$,

$\begin{array}{rcl}sec{h}^{-1}\left(\frac{1}{2}\right)& =& log\left[\frac{1+\sqrt{\left(1–{\displaystyle \frac{1}{4}}\right)}}{{\displaystyle \frac{1}{2}}}\right]\\ & =& log2\left(1+\frac{\surd 3}{2}\right)\\ & =& \mathit{l}\mathit{o}\mathit{g}\mathbf{(}\mathbf{2}\mathbf{+}\mathbf{\surd }\mathbf{3}\mathbf{)}\end{array}$

Hence, Option ‘C’ is Correct.