Question

given that alpha + beta =90 degree show that square root cos alpha cosec beta - cos alpha sin beta= sin alpha

Answer 1

$given \\ \alpha + \beta = \frac{\pi}{2} \\ \sqrt{ \cos( \alpha ) \csc( \beta ) - \cos( \alpha ) \sin( \beta ) } \\ = \sqrt{ \frac{ \cos( \alpha ) }{sin( \beta )} - \sin( \beta ) \cos( \alpha ) } \\ = \sqrt{ \frac{ \cos( \alpha ) - { \sin( \beta ) }^{2} \cos( \alpha ) }{ \sin( \beta ) } } \\ = \sqrt{ \frac{ \cos( \alpha ) (1 - { \sin( \beta ) }^{2}) }{ \sin( \beta ) } } \\ < \alpha + \beta = \frac{\pi}{2} > < \sin( \alpha ) = \cos( \beta ) > < \sin( \beta ) = \cos( \alpha ) > \\ \sqrt{ { \cos( \beta ) }^{2} } \\ = \cos( \beta ) \\ = \sin( \alpha )$
hope it helps