Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:

cot² A - cos² A = cot² A cos² A

The L.H.S of above equation can be written as,

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

Since, L.H.S. = cot² A. cos² A = R.H.S., hence proved that cot² A - cos² A = cot² A cos² A.