Prove that cosA (1 + cotA) + sinA (1 + tanA = secA + cosecA - Vidyadip

Question

Prove that
cosA (1 + cotA) + sinA (1 + tanA = secA + cosecA

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Answer

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

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