Question

Prove that
sinA/1-cosA=
cosecA+ cotA​

Answer 1

To prove:

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

Solution:

$\implies \: \frac{ \sin( \alpha ) }{ 1 - \cos( \alpha ) }$

• Rationalise the denominator , we get :

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

$\boxed{ \huge{ proved}}$

Answer 2

Step-by-step explanation:

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