Question

(cotø + cosecø)²=1+cosø/1-cosø
$pleace \: help \: me \: to \: solve \: this$
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Answer 1

$Answer \: \\ \\ GIVEN \: \: QUESTION \: \: Is \: \: \\ \\ ( \cot(x) + \csc(x) ) {}^{2} = \frac{1 + \cos(x) }{1 - \cos(x) } \\ \\ lhs \\ \\ ( \cot(x) + \csc(x) ) {}^{2} \\ \\ replace \: \: \: \: \cot(x) \: \: by \: \: \frac{ \cos(x) }{ \sin(x) } \\ and \\ \csc(x) \: \: by \: \: \frac{1}{ \sin(x) } \\ \\ ( \frac{ \cos(x) }{ \sin( x ) } + \frac{1}{ \sin(x) } ) {}^{2} \\ \\ \frac{(1 + \cos(x)) {}^{2} }{ \sin {}^{2} (x) } \\ \\ \frac{(1 + \cos(x) ) {}^{2} }{(1 - \cos {}^{2} (x)) } \\ \\ \frac{(1 + \cos(x) )\times(1 + \cos(x)) }{(1 + \cos(x) ) \times (1 - \cos(x) )} \\ \\ \frac{1 + \cos(x) }{1 - \cos(x) } \: \: hence \: proved \: \\ \\ Note\: \: \: \: \: \\ \\ 1) \: \: \: \sin {}^{2} (x) = 1 - \cos {}^{2} (x) \\ \\ 2) \: \: 1 - \cos {}^{2} (x) = (1 - \cos(x) ) \times (1 + \cos(x) ) \\ \\ here \: \: \: x \: \: = \: \: phi$

Answer 2

Step-by-step explanation:

${ (\cot( \alpha ) + \csc( \alpha ) )}^{2} \\ = { (\frac{ \cos( \alpha ) }{ \sin( \alpha ) } + \frac{1}{ \sin( \alpha ) } ) }^{2} \\ = {( \frac{1 + \cos( \alpha ) }{ \sin( \alpha ) } )}^{2} \\ = \frac{ {(1 + \cos( \alpha ) )}^{2} }{ { \sin( \alpha ) }^{2} } \\ = \frac{ {(1 + \cos( \alpha )) }^{2} }{1 - { \cos( \alpha ) }^{2} } \\ = \frac{ {(1 + \cos( \alpha ) )}^{2} }{(1 + \cos( \alpha ))(1 - \cos( \alpha ) )} \\ = \frac{1 + \cos( \alpha ) }{1 - \cos( \alpha ) }$

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