Prove the following identity:

cosec A (1 + cos A)(cosec A - cot A) = 1

Solving L.H.S. of the equation :

⇒ cosec A(1 + cos A)(cosec A - cot A)

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

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

Hence, proved that cosec A (1 + cos A)(cosec A - cot A) = 1.