Question

show that sintheta/1+costheta+1+costheta/sintheta=2cosectheta​

Answer 1

LHS:

$\frac{ \sin(θ) }{1 + \cos( \theta) } + \frac{1 + \cos( \: \theta) }{ \sin(\theta \: ) }$

$\frac{ { \sin }^{2} \theta \: + {(1 + \cos(\theta)) }^{2} }{ \sin(\theta) \times \cos(\theta) }$

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

$\frac{1 + 1 + 2cos\theta}{sin \theta \: (1 + cos \theta) }$

$\frac{2(1 + cos\theta)}{sin\theta \: (1 + cos\theta)}$

$\frac{2}{ sin \theta} \\ 2cosec \theta$

Hence proved.

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