Prove. ( A+1 A )^2+ ( A-1 A )^2=2 (1+ ^2 A 1- ^2 A )

Question

Prove.$\left(\tan A+\frac{1}{\cos A}\right)^2+\left(\tan A-\frac{1}{\cos A}\right)^2=2\left(\frac{1+\sin ^2 A}{1-\sin ^2 A}\right)$

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Answer

$\text { LHS }=\left(\tan A+\frac{1}{\cos A}\right)^2+\left(\tan A-\frac{1}{\cos A}\right)^2$
$=\left(\frac{\sin A+1}{\cos A}\right)^2+\left(\frac{\sin A-1}{\cos A}\right)^2$
$=\frac{\sin ^2 A+1+2 \sin A+\sin ^2 A+1-2 \sin A}{\cos ^2 A}$
$=\frac{2+2 \sin ^2 A}{\cos ^2 A}$
$=2\left(\frac{1+\sin ^2 A}{1-\sin ^2 A}\right)=\text { RHS }$

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