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: $\cos A+\sin A$
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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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