Question

Solve the above question

Answer 1

Step-by-step explanation:

$\frac{ \sqrt{1 - \sin( \alpha ) } }{ \sqrt{1 + \sin( \alpha ) } } = \sqrt{ \frac{1 - \sin( \alpha ) }{1 + \sin( \alpha ) } } \\ \\ = \sqrt{ \frac{1 - \sin( \alpha ) }{1 + \sin( \alpha ) } } \times \frac{ \sqrt{1 - \sin( \alpha ) } }{ \sqrt{1 - \sin( \alpha ) } } \\ \\ = \frac{1 - \sin( \alpha ) }{ \sqrt{1 - { \sin( \alpha ) }^{2} } } \\ \\ = \frac{1 - \sin( \alpha ) }{ \sqrt{ { \cos( \alpha ) }^{2} } } \\ \\ = \frac{1 - \sin( \alpha ) }{ \cos( \alpha ) } \\ \\ = \frac{1 - \sin( \alpha ) }{ \cos( \alpha ) } \times \frac{1 + \sin( \alpha ) }{1 + \sin( \alpha ) } \\ \\ = \frac{1 - { \sin( \alpha ) }^{2} }{ \cos( \alpha ) \times (1 + \sin( \alpha )) } \\ \\ = \frac{ { \cos( \alpha ) }^{2} }{ \cos( \alpha ) \times (1 + \sin( \alpha )) } \\ \\ = \frac{ \cos( \alpha ) }{1 + \sin( \alpha ) } = rhs.$