Prove the following identities :

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

Solving L.H.S. of the above equation :

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

By formula,

sin² A + cos² A = 1

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

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

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