What kind of function is f(x)=log((1+sinx)/(1-sinx))? Whether odd, even or none?

Nov 1, 2017

$f \left(x\right) = \log \left(\frac{1 + \sin x}{1 - \sin x}\right)$ is an odd function.

Explanation:

A function is even if $f \left(- x\right) = f \left(x\right)$

and it is odd if $f \left(- x\right) = - f \left(x\right)$

here we have $f \left(x\right) = \log \left(\frac{1 + \sin x}{1 - \sin x}\right) = \log \left(1 + \sin x\right) - \log \left(1 - \sin x\right)$#

Hence $f \left(- x\right) = \log \left(\frac{1 + \sin \left(- x\right)}{1 - \sin \left(- x\right)}\right)$

= $\log \left(\frac{1 - \sin x}{1 + \sin x}\right)$ - (as $\sin \left(- x\right) = - \sin x$)

= $\log \left(1 - \sin x\right) - \log \left(1 + \sin x\right)$

= $- f \left(x\right)$

Hence $f \left(x\right) = \log \left(\frac{1 + \sin x}{1 - \sin x}\right)$ is an odd function.