Question

prove that (cosecA-cotA)²= (-cosA/1+cosA)
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Answer 1

To prove :

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

LHS :

$\implies \: {( \csc( \alpha ) - \cot( \alpha ) )}^{2} \\ \\ \implies (\dfrac{1}{ \sin( \alpha ) } - \dfrac{ \cos( \alpha ) }{ \sin( \alpha ) } ) {}^{2} \\ \\ \implies \: {( \dfrac{1 - \cos( \alpha ) }{ \sin( \alpha ) } })^{2} \\ \\ \implies \dfrac{ {(1 - \cos( \alpha )) }^{2} }{ { \sin }^{2} \alpha } \\ \\ \implies \: \dfrac{ {(1 - \cos( \alpha )) }^{2} }{1 - { \cos }^{2} \alpha } \\ \\ \implies \dfrac{ {(1 - \cos( \alpha )) }^{ \cancel{2}} }{(1 + \cos( \alpha )) ( \cancel{1 - \cos( \alpha ))} } \\ \\ \implies \: \frac{1 - \cos( \alpha ) }{1 + \cos( \alpha ) }$

LHS = RHS

Proved .

Formula :

$\star \: { \sin}^{2} \theta + { \cos}^{2} \theta = 1 \\ \\ \star \: (x + y)(x - y) = {x}^{2} - {y}^{2}$

Answer 2

Answer:

• Please refer to the attachment ✔️