Question

If $\tan \left(\frac{x}{2}\right).\coth \left(\frac{x}{2}\right)=1,$ then the value of $\cos x.\cosh x$ is

• A. 1
• B. $-1$
• C. $co{s}^{2}x$
• D. $sin{h}^{2}x$

Answer 1

Answer: (A)

We have,
$\tan \frac{x}{2}=\tanh =\frac{x}{2}$
$\Rightarrow$ $\frac{{\tan }^{2}x/2}{1}=\frac{{\tanh }^{2}x/2}{1}$
$\Rightarrow$ $\frac{1+{\tan }^{2}x/2}{1-{\tan }^{2}x/2}=\frac{1+{\tanh }^{2}x/2}{1-{\tanh }^{2}x/2}$
(by using componendo and dividendo rule)
$\Rightarrow$ $\frac{1}{\cos x}=\cosh x$
$\Rightarrow$ $\cos x.\cosh x=1$