Simplify: cosecθ(1 – cosθ)(cosecθ + cotθ)

Correct option is D

​$$\textbf{Given:} \\\text{Expression } = \cosec\theta (1 - \cos\theta)(\cosec\theta + \cot\theta) \\\\textbf{Formula Used:} \\\cosec\theta = \dfrac{1}{\sin\theta}, \quad\cot\theta = \dfrac{\cos\theta}{\sin\theta} \\1 - \cos\theta = \dfrac{(1 - \cos\theta)(1 + \cos\theta)}{1 + \cos\theta}= \dfrac{1 - \cos^2\theta}{1 + \cos\theta}= \dfrac{\sin^2\theta}{1 + \cos\theta}\\\textbf{Solution:} \\\cosec\theta (1 - \cos\theta)(\cosec\theta + \cot\theta) \\= \dfrac{1}{\sin\theta} (1 - \cos\theta)\left( \dfrac{1}{\sin\theta} + \dfrac{\cos\theta}{\sin\theta} \right) \\= \dfrac{1}{\sin\theta} (1 - \cos\theta)\left( \dfrac{1 + \cos\theta}{\sin\theta} \right) \\= \dfrac{(1 - \cos\theta)(1 + \cos\theta)}{\sin^2\theta} \\= \dfrac{1 - \cos^2\theta}{\sin^2\theta} \\= \dfrac{\sin^2\theta}{\sin^2\theta} \\= 1$$​​