(a) Prove that: (tan θ)/(1+ tan^2 θ)+(cot θ)/(1+ cot^2 θ)=2 sin θ cos θ.

L.H.S. \(\displaystyle =\frac{\tan \theta}{\sec ^{2} \theta}+\frac{\cot \theta}{\operatorname{cosec}^{2} \theta}\)

\[=\frac{\sin \theta}{\cos \theta} \times \cos ^{2} \theta+\frac{\cos \theta}{\sin \theta} \times \sin ^{2} \theta \]

\[=2 \sin \theta \cos \theta=\text { R.H.S. } \]

\[\begin{aligned} & =\frac{1-2 \times\left(\dfrac{1}{\sqrt{3}}\right)^{2}-(\sqrt{2})^{2}}{\left(\dfrac{\sqrt{3}}{2}\right)^{2}} \\ & =\frac{-20}{9} \end{aligned} \]