Prove the following identities :

(1 + cot A - cosec A)(1 + tan A + sec A) = 2

Solving L.H.S. of the equation :

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

By formula,

sin² A + cos² A = 1.

$\Rightarrow \dfrac{1 + \text{2 cos A sin A - 1}}{\text{sin A cos A}} \\[1em] \Rightarrow \dfrac{\text{2 cos A sin A}}{\text{cos A sin A}} \\[1em] \Rightarrow 2.$

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

Hence, proved that (1 + cot A - cosec A)(1 + tan A + sec A) = 2.