For the curve $\mathrm{y}=3\sin \mathrm{\theta cos}\mathrm{\theta },\mathrm{x}={\mathrm{e}}^{\mathrm{\theta }}\sin \mathrm{\theta },0\le \mathrm{\theta }\le \mathrm{\pi }\text{; }$ 

If  tangent at a point  is parallel to x-axis, then $\theta =$

Given $\mathrm{y}=3\sin \mathrm{\theta cos}\mathrm{\theta },\mathrm{x}={\mathrm{e}}^{\mathrm{\theta }}\sin \mathrm{\theta }$ For tangent parallel to x axis, we should have 
$\frac{\mathrm{dy}}{\mathrm{dx}}=0\Rightarrow \frac{\mathrm{dy}}{\mathrm{d\theta }}=0\text{ and }\frac{\mathrm{dx}}{\mathrm{d\theta }}\ne 0\Rightarrow \frac{\mathrm{d}}{\mathrm{d\theta }}\left(3\frac{\sin 2\mathrm{\theta }}{2}\right)=0\Rightarrow 3\cos 2\mathrm{\theta }=0\Rightarrow 2\mathrm{\theta }=\frac{\mathrm{\pi }}{2}$ 
$\Rightarrow \mathrm{\theta }=\frac{\mathrm{\pi }}{4}$ Note that for $\mathrm{\theta }=\frac{\mathrm{\pi }}{4},\frac{\mathrm{dx}}{\mathrm{d\theta }}={\mathrm{e}}^{\mathrm{\theta }}[\sin \mathrm{\theta }+\cos \mathrm{\theta }]\ne 0$.