Prove. 1+ A A + A 1+ A =2 A - Vidyadip

Question

Prove.
$\frac{1+\sin A}{\cos A}+\frac{\cos A}{1+\sin A}=2 \sec A$

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Answer

$\text { LHS }=\frac{1+\sin A}{\cos A}+\frac{\cos A}{1+\sin A}$
$ =\frac{(1+\sin A)^2+\cos ^2 A}{\cos A(1+\sin A)}$
$=\frac{1+\sin ^2 A+2 \sin A+\cos ^2 A}{\cos A(1+\sin A)} $
$=\frac{1+2 \sin A+1}{\cos A(1+\sin A)} $
$=\frac{2(1+\sin A)}{\cos A(1+\sin A)} $
$ =2 \sec A=\text { RHS }$

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