Question

1+cosec theta -cot theta /1+cosec theta +cot theta =?​

Answer 1

Step-by-step explanation:

Given that,

$\frac{1 + \: cosec \: \alpha \: - \: \cot \:\alpha }{1 + \: cosec \: \alpha \: + \: \cot \: \alpha }$

Since ,

${ \cosec^{2} \alpha } - {cot}^{2} \alpha = 1$

$= \frac{( {cosec}^{2} \alpha - {cot}^{2} \alpha ) + (cosec \: \alpha - \: cot \: \alpha )}{1 + (cosec \: \alpha + \: cot \: \alpha )}$

$= \frac{( {cosec} \alpha + {cot} \alpha ) (cosec \: \alpha - cot \: \alpha ) + (cosec \: \alpha - \: cot \: \alpha )}{1+ (cosec \: \alpha + \: cot \: \alpha )}$

$= \frac{(cosec \: \alpha - cot \alpha )(cosec \: \alpha + \: cot \: \alpha + 1)}{1 + \: cosec \: \alpha \: + \: cot \: \alpha }$

$= cosec \: \alpha - \: cot \: \alpha$

$= \frac{1}{ \sin \alpha } - \frac{ \cos\alpha }{ \sin \alpha }$

$= \frac{1 - \cos \alpha }{ \sin \alpha }$

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