Question

The value of $sech\left(i\mathrm{\pi }\right)$ is

• A. $-1$
• B. $i$
• C. $0$
• D. $1$

Answer 1

The correct option is A

$-1$

Explanation for the correct answer:

Compute the required value:

we know that,

$\cos h\left(x\right)=\frac{e^x+e^{-x}}{2}$

$\Rightarrow$ $sech\left(x\right)=\frac{2}{e^x+e^{-x}}$

Substitute the value of $x=i\mathrm{\pi }$ we get,

$\Rightarrow$ $sech\left(i\mathrm{\pi }\right)=\frac{2}{e^{i\mathrm{\pi }}+e^{-i\mathrm{\pi }}}$

$\Rightarrow$ $sech\left(i\mathrm{\pi }\right)=\frac{2}{\mathrm{cos\pi }+i\mathrm{sin\pi }+\mathrm{cos\pi }-i\mathrm{sin\pi }}$ $...[\because e^{i\theta }=\cos \theta +isin\theta ]$

$\Rightarrow$ $sech\left(i\mathrm{\pi }\right)=\frac{2}{2\cos \mathrm{\pi }}$

$\Rightarrow$ $sech\left(i\mathrm{\pi }\right)=sec\pi =-1$

Hence, the value of $sech\left(i\mathrm{\pi }\right)$ is $-1$.

Hence, option $\left(A\right)$ is the correct answer.