sin C / 1 + cos C + sin C / 1 - cos C

= sin c /(1+cos c) + sin c /(1-cos c)
= sin c(1-cos c) + sin c(1+cos c) / (1+cos c)(1-cos c)
= (sin c + sin c)/(1-cos²c)
= 2 sin c / sin²c
= 2/sin c
= 2cosec C