Question

Prove ;
$\frac{\tan A}{(1+{\tan }^{2}A{)}^2}+\frac{\cot A}{(1+{\cot }^{2}A{)}^2}=\sin A\cos A$

Answer 1

Taking L.H.S.-

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

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

$=\frac{\left(\frac{\sin A}{\cos A}\right)}{\left(\frac{1}{{\cos }^{4}A}\right)}+\frac{\left(\frac{\cos A}{\sin A}\right)}{\left(\frac{1}{{\sin }^{4}A}\right)}$

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

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

$=\sin A\cos A$