Prove that: 1 1+ (90^ -A ) +1 1- (90^ -A ) =2 ^2 (90^ -A ) - Vidyadip

Question

Prove that:
$\frac{1}{1+\sin \left(90^{\circ}-A\right)}+\frac{1}{1-\sin \left(90^{\circ}-A\right)}=2 \sec ^2\left(90^{\circ}-A\right)$

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Answer

$\frac{1}{1+\sin \left(90^{\circ}-A\right)}+\frac{1}{1-\sin \left(90^{\circ}-A\right)}$
$=\frac{1}{1+\cos A}+\frac{1}{1-\cos A}$
$=\frac{1-\cos A+1+\cos A}{(1+\cos A)(1-\cos A)}$
$=\frac{2}{1-\cos ^2 A}$
$=2 \operatorname{cosec}^2 A $
$ =2 \sec ^2\left(90^{\circ}-A\right)$

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