Question

Prove that following functions are increasing in given intervals.
$f\left(x\right)=log\sin x,x\in (0,\frac{\pi }{2})$.

Answer 1

$f\left(x\right)=\frac{1\times \cos x}{\sin x}$

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

$forincrea\sin gf\left(x\right)>0$

$\Rightarrow \therefore \cot x>0$

$\Rightarrow \frac{\cos x}{\sin x}>0$

$\Rightarrow \cos x>0,x>0$

$\Rightarrow \sin x>0,x>\frac{\pi }{2}$

$\therefore f\left(x\right)=\log \sin xnincrea\sin g\left(0,\frac{\pi }{2}\right)$