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Prove that : $\left(\frac{1}{\cos \text{A}}-\cos \text{A}\right) \left(\frac{1}{\sin \text{A}}-\sin \text{A}\right) = \frac{1}{\tan \text{A} + \cot \text{A}}$
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LHS $$\begin{aligned}& = \left(\frac{1-\cos^2 \text{A}}{\cos \text{A}}\right) \left(\frac{1-\sin^2 \text{A}}{\sin \text{A}}\right) \\ & = \frac{\sin^2 \text{A} \cos^2 \text{A}}{\cos \text{A} \sin \text{A}} \\ & = \sin \text{A} \cdot \cos \text{A} \\ & \text{RHS} = \frac{1}{\frac{\sin \text{A}}{\cos \text{A}} + \frac{\cos \text{A}}{\sin \text{A}}} \\ & = \frac{1}{\frac{\sin^2 \text{A} + \cos^2 \text{A}}{\sin \text{A} \cos \text{A}}} \\ & = \sin \text{A} \cdot \cos \text{A} \\ & \therefore \text{LHS} = \text{RHS}\end{aligned}$$