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

Question

Prove.
$\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\sin A+\cos A$

✓

Answer

$\text { LHS }=\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A} $
$ =\frac{\cos A}{1-\frac{\sin A}{\cos A}}+\frac{\sin A}{1-\frac{\cos A}{\sin A}}=\frac{\cos A}{\frac{\cos A-\sin A}{\cos A}}+\frac{\sin A}{\frac{\sin A-\cos A}{\sin A}} $
$ \frac{\cos ^2 A}{\cos A-\sin A}+\frac{\sin ^2 A}{\sin A-\cos A}=\frac{\cos ^2 A-\sin ^2 A}{\cos A-\sin A} $
$ =\sin A+\cos A=\text { RHS }$

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