Prove the following trigonometric identities:$ \frac{\sin \theta}{1-\cos \theta}=\operatorname{cosec} \theta+\cot \theta $

To do:

We have to prove that \( \frac{\sin \theta}{1-\cos \theta}=\operatorname{cosec} \theta+\cot \theta \).

Solution:

We know that,

$\sin ^{2} \theta+cos ^{2} \theta=1$.......(i)

$\operatorname{cosec} \theta=\frac{1}{\sin \theta}$........(ii)

$\cot \theta=\frac{\cos \theta}{\sin \theta}$........(iii)

Therefore,

$\frac{\sin \theta}{1-\cos \theta}=\frac{\sin \theta}{1-\cos \theta}\times \frac{1+\cos \theta}{1+\cos \theta}$     (Multiply and divide by $1+\cos \theta$)

$=\frac{(\sin \theta)(1+\cos \theta)}{(1-\cos \theta)(1+\cos \theta)}$

$=\frac{\sin \theta(1+\cos \theta)}{1^2-\cos^2 \theta)}$

$=\frac{\sin \theta(1+\cos \theta)}{\sin^2 \theta}$      (From (i))

$=\frac{1+\cos \theta}{\sin \theta}$           

$=\frac{1}{\sin \theta}+\frac{\cos \theta}{\sin \theta}$

$=\operatorname{cosec} \theta+\cot \theta$          (From (ii) and (iii))

Hence proved.   

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Updated on: 10-Oct-2022

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