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