Trigonometric identity proof

LHS $= \frac{\tan A - \cot A}{\sin A \cos A} = \frac{\tan A}{\sin A \cos A} - \frac{\cot A}{\sin A \cos A}$ Substitute $\tan A = \frac{\sin A}{\cos A}$ and $\cot A = \frac{\cos A}{\sin A}$: $\frac{\sin A / \cos A}{\sin A \cos A} = \frac{1}{\cos^2 A} = \sec^2 A$ $\frac{\cos A / \sin A}{\sin A \cos A} = \frac{1}{\sin^2 A} = \text{cosec}^2 A$ LHS $= \sec^2 A - \text{cosec}^2 A = (1 + \tan^2 A) - (1 + \cot^2 A) = \tan^2 A - \cot^2 A = \text{RHS}$.