Prove that A (1 + A) + A (1 + A = A + cosec A) - Vidyadip

Question

Prove that
$\cos A (1 + \cot A) + \sin A (1 + \tan A = \sec A + \operatorname{cosec} A)$

✓

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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