Prove that :

tan A + cot A = sec A. cosec A.

To prove: tan A + cot A = sec A. cosec A.

Solving L.H.S. of the equation :

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

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

Hence, proved that tan A + cot A = sec A. cosec A.