Prove the following identity : cosecθ(1 + cosθ)(cosecθ - cotθ) = … - Vidyadip

Question

Prove the following identity :
cosecθ(1 + cosθ)(cosecθ - cotθ) = 1

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Answer

$
\begin{aligned}
& \text { LHS }=\operatorname{cosec} \theta(1+\cos \theta)(\operatorname{cosec} \theta-\cot \theta) \\
& =\frac{1}{\sin \theta}(1+\cos \theta)\left(\frac{1}{\sin \theta}-\frac{\cos \theta}{\sin \theta}\right) \\
& =\frac{(1+\cos \theta)}{\sin \theta}\left(\frac{1-\cos \theta}{\sin \theta}\right) \\
& =\frac{1-\cos ^2 \theta}{\sin ^2 \theta}=\frac{\sin ^2 \theta}{\sin ^2 \theta}=1=\text { RHS }
\end{aligned}
$

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