Prove the following identity:

cot² A - cos² A = cos² A cot² A

Solving L.H.S of equation,

$\Rightarrow \dfrac{\cos^2 A}{\sin^2 A} - \cos^2 A \\[1em] \Rightarrow \cos^2 A \Big( \dfrac{1}{\sin^2 A} - 1 \Big) \\[1em] \Rightarrow \cos^2 A \Big( \dfrac{1 - \sin^2 A}{\sin^2 A} \Big) \\[1em] \Rightarrow \cos^2 A \Big(\dfrac{\cos^2 A}{\sin^2 A} \Big) \\[1em] \Rightarrow \cos^2 A \cot^2 A.$

Since, L.H.S. = R.H.S.,

Hence, proved that cot² A - cos² A = cos² A cot² A.