Question

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

Answer 1

Step-by-step explanation:

We have,

$\rm \frac{1 + \cosec( \theta) + \cot( \theta) }{1 + \cosec( \theta) - \cot( \theta) }$

$\rm = \frac{1 + \cosec( \theta) + \cot( \theta) }{ \cosec ^{2} ( \theta) - \cot ^{2} ( \theta) + \cosec( \theta) - \cot( \theta) }$

$\rm = \frac{1 + \cosec( \theta) + \cot( \theta) }{ \{\cosec ( \theta) - \cot ( \theta) \} \{ \cosec( \theta) + \cot( \theta) \} + \cosec( \theta) - \cot( \theta) }$

$\rm = \frac{1 + \cosec( \theta) + \cot( \theta) }{ \{\cosec ( \theta) - \cot ( \theta) \} \{ \cosec( \theta) + \cot( \theta) + 1 \} }$

$\rm = \frac{1 }{\cosec( \theta) - \cot( \theta) }$

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

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

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

$\rm = \frac{ 2\sin \bigg( \dfrac{\theta}{2} \bigg) \cos \bigg( \dfrac{\theta}{2} \bigg) }{ 2\sin^{2} \bigg( \dfrac{\theta}{2} \bigg) }$

$\rm = \frac{ \cos \bigg( \dfrac{\theta}{2} \bigg) }{ \sin \bigg( \dfrac{\theta}{2} \bigg) }$

$\rm = \cot \bigg( \dfrac{\theta}{2} \bigg)$