Question
Prove that:
$\frac{\tan A}{\left(1+\tan ^2 A\right)^2}+\frac{\cot A}{\left(1+\cot ^2 A\right)^2}=\sin A \cos A$
$\frac{\tan A}{\left(1+\tan ^2 A\right)^2}+\frac{\cot A}{\left(1+\cot ^2 A\right)^2}=\sin A \cos A$
✓
Answer
Taking RHS
$\frac{\tan A }{\left(1+\tan ^2 A\right)^2}+\frac{\cot A }{\left(1+\cot ^2 A\right)^2}$
$=\frac{\tan A }{\left(\sec ^2 A\right)^2}+\frac{\cot A }{\left(\operatorname{cosec}^2 A\right)^2}$
$=\frac{\tan A }{\sec ^4 A}+\frac{\cot A }{\operatorname{cosec}^4 A}$
$=\frac{\sin A }{\cos A } \cdot \cos ^4 A+\frac{\cos A }{\sin A } \cdot \sin ^4 A$
$=\sin A \cos ^3 A+\cos A \sin ^3 A$
$=\sin A \cos A\left(\cos ^2 A+\sin ^2 A\right)\left[A s, \sin ^2 \theta+\cos ^2 \theta=1\right]$
$=\sin A \cos A$
$=\text { RHS }$
Proved!
$\frac{\tan A }{\left(1+\tan ^2 A\right)^2}+\frac{\cot A }{\left(1+\cot ^2 A\right)^2}$
$=\frac{\tan A }{\left(\sec ^2 A\right)^2}+\frac{\cot A }{\left(\operatorname{cosec}^2 A\right)^2}$
$=\frac{\tan A }{\sec ^4 A}+\frac{\cot A }{\operatorname{cosec}^4 A}$
$=\frac{\sin A }{\cos A } \cdot \cos ^4 A+\frac{\cos A }{\sin A } \cdot \sin ^4 A$
$=\sin A \cos ^3 A+\cos A \sin ^3 A$
$=\sin A \cos A\left(\cos ^2 A+\sin ^2 A\right)\left[A s, \sin ^2 \theta+\cos ^2 \theta=1\right]$
$=\sin A \cos A$
$=\text { RHS }$
Proved!
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