Question

if x=1+sin theta /cos theta prove that cosec theta =x^2+1/x^2-1​

Answer 1

$x = 1 + \frac{ \sin( \alpha ) }{ \cos( \alpha ) } \\ x = 1 + \tan( \alpha ) \\ x - 1 = \frac{1}{ \cot( \alpha ) } \\ squaring \: on \: both \: sides \\ ( {x - 1)}^{2} = \frac{1}{ { \cot( \alpha ) }^{2} } \\ \frac{1}{ {(x - 1)}^{2} } = { \cot( \alpha ) }^{2} \\ adding \: one \: on \: bith \: sides \\ \frac{1 + {(x - 1)}^{2} }{ {(x - 1)}^{2} } = { \csc( \alpha ) }^{2} \\ \csc( \alpha ) = \frac{ {x}^{2} + 1}{ {x}^{2} - 1 }$

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