Prove that: $\left(\frac{1}{\cos A} - \cos A ight)\left(\frac{1}{\sin A} - \sin A ight) = \frac{1}{ an A + \cot A}$

Question

Prove that: $\left(\frac{1}{\cos A} - \cos Aight)\left(\frac{1}{\sin A} - \sin Aight) = \frac{1}{	an A + \cot A}$

Accepted Answer

Step 1: LHS $= \left(\frac{1 - \cos^2 A}{\cos A}ight)\left(\frac{1 - \sin^2 A}{\sin A}ight)$
Step 2: $= \frac{\sin^2 A}{\cos A} \cdot \frac{\cos^2 A}{\sin A} = \sin A \cdot \cos A$
Step 3: RHS $= \frac{1}{	an A + \cot A} = \frac{1}{\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}} = \frac{\sin A \cos A}{\sin^2 A + \cos^2 A} = \sin A \cos A$
$	herefore$ LHS $=$ RHS

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