Question

Prove that:
$\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\cos A+sinA$

Answer 1

To prove:- $\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\cos A+\sin A$

Proof:-

Taking L.H.S., we have

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

$=\frac{\cos A}{1-\frac{\sin A}{\cos A}}+\frac{\sin A}{1-\frac{\cos A}{\sin A}}$

$=\frac{{\cos }^{2}A}{\cos A-\sin A}+\frac{{\sin }^{2}A}{\sin A-\cos A}$

$=\frac{{\cos }^{2}A}{\cos A-\sin A}-\frac{{\sin }^{2}A}{\cos A-\sin A}$

$=\frac{{\cos }^{2}A-{\sin }^{2}A}{\cos A-\sin A}$

$=\frac{(\cos A+\sin A)(\cos A-\sin A)}{\cos A-\sin A}$

$=\cos A+\sin A$

$=$ R.H.S.

Hence proved.