Prove the following identity:

sec A (1 - sin A)(sec A + tan A) = 1

Solving L.H.S. of the equation :

⇒ sec A(1 - sin A)(sec A + tan A)

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

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

Hence, proved that sec A (1 - sin A)(sec A + tan A) = 1.