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Given:- Function $f(x)=\log \sin x$

Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$.

(i) If $f^{\prime}(x)>0$ for all $x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$

(ii) If $f^{\prime}(x)<0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$

Algorithm:-

(i) Obtain the function and put it equal to $f(x)$

(ii) Find $f^{\prime}(x)$

(iii) Put $f^{\prime}(x)>0$ and solve this inequation.

For the value of $x$ obtained in (ii) $f(x)$ is increasing and for remaining points in its domain it is decreasing.

Here we have,

$\Rightarrow \mathrm{f}(\mathrm{x})=\frac{\mathrm{d}}{\mathrm{dx}}(\log \sin \mathrm{x})$

$\Rightarrow \mathrm{f}(\mathrm{x})=\frac{1}{\sin \mathrm{x}} \times \cos \mathrm{x}$

$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\cot (\mathrm{x})$

Taking different region from 0 to $\pi$

a) let $x \in\left(0, \frac{\pi}{2}\right)$

$\Rightarrow \cot (x)>0$

$\Rightarrow f^{\prime}(x)>0$

Thus $f(x)$ is increasing in $\left(0, \frac{\pi}{2}\right)$

b) $\operatorname{let} \mathrm{x} \in\left(\frac{\pi}{2}, \pi\right)$

$\Rightarrow \cot (x)<0$

$\Rightarrow f^{\prime}(x)<0$

Thus $f(x)$ is decreasing in $\left(\frac{\pi}{2}, \pi\right)$

Hence proved