AMATHS!~ HELP PROVE IDENTITY cosec x + cot x = cot(x/2)

Prove that cosec x + cot x = cot(x/2)

LHS=1/sinx +cosx/sinx
=(1+cosx)/sinx
= 2cos^2(x/2)/2sin(x/2)cos(x/2)
=cos(x/2)/sin(x/2)
= cot(x/2)