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

tan A + cot A = sec A cosec A

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

$\Rightarrow \dfrac{\text{sin A}}{\text{cos A}} + \dfrac{\text{cos A}}{\text{sin A}} \\[1em] \Rightarrow \dfrac{\text{sin}^2 A + \text{cos}^2 A}{\text{cos A.sin A}} \\[1em] \Rightarrow \dfrac{1}{\text{cos A.sin A}} \\[1em] \Rightarrow \dfrac{1}{\text{cos A}} \times \dfrac{1}{\text{sin A}} \\[1em] \Rightarrow \text{sec A. cosec A}.$

Since, L.H.S. = sec A.cosec A = R.H.S., hence proved that tan A + cot A = sec A cosec A.