LHS,
[tex] = \frac{ \sin(x)(1 + \cos(x)) + \sin(x)(1 - \cos(x) ) }{(1 + \cos(x))(1 - \cos(x)) } [/tex]
[tex] = \frac{ \sin(x) + \sin(x) \cos(x) + \sin(x) - \sin(x) \cos(x) }{1 - { (\cos(x)) }^{2} } [/tex]
[tex] = \frac{2 \sin(x) }{( { \sin(x) )}^{2} } [/tex]
[tex] = \frac{2}{ \sin(x) } [/tex]
[tex] = \csc(x) [/tex]
hence, proved.
