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}}\text{ }x $
• D. $ sin{{h}^{2}}\text{ }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 $