Question

The value of $\frac{(e^{2\theta }-1)}{(e^{2\theta }+1)}$ is

• A. $coth\theta$
• B. $coth2\theta$
• C. $\tan h\theta$
• D. $\tan h2\theta$

Answer 1

The correct option is C

$\tan h\theta$

Explanation for the correct option:

Find the required value.

Given: $\frac{(e^{2\theta }-1)}{(e^{2\theta }+1)}$

$$\begin{gathered} \frac{(e^{2\theta }-1)}{(e^{2\theta }+1)}=\frac{(e^{\theta }{e}^{\theta }–1)}{(e^{\theta }{e}^{\theta }+1)} \\ =\frac{e^{\theta }\left(e^{\theta }–{\displaystyle \frac{1}{e^{\theta }}}\right)}{e^{\theta }\left(e^{\theta }+\frac{1}{e^{\theta }}\right)} \\ =\frac{\left(e^{\theta }–e^{-\theta }\right)}{\left(e^{\theta }+e^{-\theta }\right)} \\ =\tan h\theta \left[\because \tan hx=\frac{\left(e^x–e^{-x}\right)}{\left(e^x+e^{-x}\right)}\right] \end{gathered}$$

Hence, option (C) is the correct answer.