Question
Prove the following identity:
$
(1-\tan A)^2+(1+\tan A)^2=2 \sec ^2 A
$
$
(1-\tan A)^2+(1+\tan A)^2=2 \sec ^2 A
$
✓
Answer
$
\begin{aligned}
& \text { LHS }=(1-\tan A)^2+(1+\tan A)^2 \\
& =1+\tan ^2 A-2 \tan A+1+\tan ^2 A+2 \tan A \\
& =2\left(1+\tan ^2 A\right)=2 \sec ^2 A=\text { RHS }
\end{aligned}
$
\begin{aligned}
& \text { LHS }=(1-\tan A)^2+(1+\tan A)^2 \\
& =1+\tan ^2 A-2 \tan A+1+\tan ^2 A+2 \tan A \\
& =2\left(1+\tan ^2 A\right)=2 \sec ^2 A=\text { RHS }
\end{aligned}
$
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