1+cosx/2sinx + sinx/2+2cosx

If that is

(1 +cosX)/(2sinX) +(sinX)/(2 +2cosX),

then,

Combine the two fractions into one fraction only.

The common denominator is 2(sinX)(1+cosX),

= [(1+cosX)^2 +(sinX)^2] / [2(sinX)(1+cosX)]

= [1 +2cosX +cos^2(X) +sin^2(X)] / [2(sinX)(1+cosX)]

= [1 +2cosX +1] / [2(sinX)(1+cosX)]

= [2 +2cosX] / [(sinX)(2 +2cosX)]

= 1 / sinX

= cscX ------------------answer.