Simplify the expression: $$\sqrt{\frac{1 + cos\theta}{1 - cos\theta}}$$ is:​

Correct option is D

Given:

​$$\sqrt{\frac{1 + cos\theta}{1 - cos\theta}}$$  

Formula Used:

​$$cosec^2\theta - cot^2\theta = 1$$​​

Solution:

​$$\sqrt{\frac{1 + cos\theta}{1 - cos\theta}}\\$$ 

Rationalize the above given terms,

​$$\begin{aligned}&\sqrt{\frac{1 + \cos\theta}{1 - \cos\theta}} \times \sqrt{\frac{1 - \cos\theta}{1 - \cos\theta}} \\&= \sqrt{\frac{(1 + \cos\theta)(1 - \cos\theta)}{(1 - \cos\theta)(1 - \cos\theta)}} \\&= \sqrt{\frac{1 - \cos^2\theta}{(1 - \cos\theta)^2}} \\&= \sqrt{\frac{\sin^2\theta}{(1 - \cos\theta)^2}} \\&= \frac{\sin\theta}{1 - \cos\theta}\end{aligned}$$

Divided by sin $$\theta$$ 

​$$= \frac{\sin\theta}{1 - \cos\theta}\\= \frac{1}{\frac{1}{\sin\theta} - \frac{\cos\theta}{\sin\theta}} \\= \frac{1}{cosec\theta - cot\theta}$$ 

we can write 1 as $$cosec^2\theta - cot^2\theta = 1$$ 

​$$\begin{aligned}&= \frac{\csc^2\theta - \cot^2\theta}{\csc\theta - \cot\theta} \\&= \frac{(\csc\theta - \cot\theta)(\csc\theta + \cot\theta)}{\csc\theta - \cot\theta} \\&= \csc\theta + \cot\theta\end{aligned}$$​​

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