Prove the following trigonometric identities.

We have to prove $\frac{\tan A}{\left(1+\tan ^{2} A\right)^{2}}+\frac{\cot A}{\left(1+\cot ^{2} A\right)^{2}}=\sin A \cos A$

We know that, $\sin ^{2} A+\cos ^{2} A=1$.

So,

$\frac{\tan A}{\left(1+\tan ^{2} A\right)^{2}}+\frac{\cot A}{\left(1+\cot ^{2} A\right)^{2}}$

$=\frac{\tan A}{\left(\sec ^{2} A\right)^{2}}+\frac{\cot A}{\left(\operatorname{cosec}^{2} A\right)^{2}}$

$=\frac{\tan A}{\sec ^{4} A}+\frac{\cot A}{\operatorname{cosec}^{4} A}$

$=\frac{\frac{\sin A}{\cos A}}{\frac{1}{\cos ^{4} A}}+\frac{\frac{\cos A}{\sin A}}{\frac{1}{\sin ^{4} A}}$

$=\frac{\sin A \cos ^{4} A}{\cos A}+\frac{\cos A \sin ^{4} A}{\sin A}$

$=\sin A \cos ^{3} A+\cos A \sin ^{3} A$

$=\sin A \cos A\left(\cos ^{2} A+\sin ^{2} A\right)$

$=\sin A \cos A$

Hence proved.