Question : $\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=$___________.
Option 1: $\tan A - \cot A$
Option 2: $\tan A + \cot A$
Option 3: $\sin A - \cos A$
Option 4: $\sin A + \cos A$
Correct Answer: $\sin A + \cos A$
Solution :
$\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}$
$= \frac{\cos A}{1-\frac{\sin A}{\cos A}} + \frac{\sin A }{1-\frac{\cos A}{\sin A}}$
$= \frac{\cos^2 A}{\cos A - \sin A}+\frac{\sin^2 A }{\sin A - \cos A}$
$= \frac{\cos^2 A}{\cos A - \sin A} - \frac{\sin^2 A}{\cos A - \sin A}$
$= \frac{(\cos A - \sin A)(\cos A + \sin A)}{\cos A - \sin A}$
$= \cos A + \sin A$
Hence, the correct answer is $\sin A + \cos A$.
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