Question
prove. $sec^2A \ cosec^2A = \tan^2A + \cot^2A + 2$
✓
Answer
$\sec ^2 A \operatorname{cosec}^2 A$
$=\left(1+\tan ^2 A\right)\left(1+\cot ^2 A\right)$
$=1+\cot ^2 A+\tan ^2 A+\tan ^2 A \cdot \cot ^2 A$
$=1+\cot ^2 A+\tan ^2 A+\frac{1}{\cot ^2 A} \times \cot ^2 A$
$=\tan ^2 A+\cot ^2 A+2$
$=\left(1+\tan ^2 A\right)\left(1+\cot ^2 A\right)$
$=1+\cot ^2 A+\tan ^2 A+\tan ^2 A \cdot \cot ^2 A$
$=1+\cot ^2 A+\tan ^2 A+\frac{1}{\cot ^2 A} \times \cot ^2 A$
$=\tan ^2 A+\cot ^2 A+2$
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