Prove the following identities :

$\dfrac{1}{\text{tan A + cot A}} = \text{cos A sin A}$

Solving L.H.S. of the equation :

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

By formula,

sin² A + cos² A = 1

$\Rightarrow \text{sin A cos A}.$

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

Hence, proved that $\dfrac{1}{\text{tan A + cot A}} = \text{cos A sin A}$.