frac{cot theta - operatorname{cosec} theta + | vedclass

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Question

(C) Given expression: $\frac{\cot 	heta - \operatorname{cosec} 	heta + 1}{\cot 	heta + \operatorname{cosec} 	heta - 1}$We know that $1 = \operator

Vedclass · Accepted Answer

$\operatorname{cosec} 	heta - \cot 	heta$

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(C) Given expression: $\frac{\cot 	heta - \operatorname{cosec} 	heta + 1}{\cot 	heta + \operatorname{cosec} 	heta - 1}$We know that $1 = \operatorname{cosec}^2 	heta - \cot^2 	heta$.Substituting this in the numerator:$= \frac{(\cot 	heta - \operatorname{cosec} 	heta) + (\operatorname{cosec}^2 	heta - \cot^2 	heta)}{\cot 	heta + \operatorname{cosec} 	heta - 1}$$= \frac{(\cot 	heta - \operatorname{cosec} 	heta) + (\operatorname{cosec} 	heta - \cot 	heta)(\operatorname{cosec} 	heta + \cot 	heta)}{\cot 	heta + \operatorname{cosec} 	heta - 1}$Taking $(\operatorname{cosec} 	heta - \cot 	heta)$ common from the numerator:$= \frac{-(\operatorname{cosec} 	heta - \cot 	heta) + (\operatorname{cosec} 	heta - \cot 	heta)(\operatorname{cosec} 	heta + \cot 	heta)}{\cot 	heta + \operatorname{cosec} 	heta - 1}$$= \frac{(\operatorname{cosec} 	heta - \cot 	heta) [(\operatorname{cosec} 	heta + \cot 	heta) - 1]}{\cot 	heta + \operatorname{cosec} 	heta - 1}$Since the term $(\cot 	heta + \operatorname{cosec} 	heta - 1)$ is common in both numerator and denominator,they cancel out:$= \operatorname{cosec} 	heta - \cot 	heta$

$\frac{\cot \theta - \operatorname{cosec} \theta + 1}{\cot \theta + \operatorname{cosec} \theta - 1}$ is equal to

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