sin2x-sinx/cos2x+cosx = sinx/1+cosx

Question

prove the identity using sum to product formulas

Answer 1

The LHS is 2cos(3x/2)sin(x/2)/2cos(3x/2)cos(x/2) = tan x/2

tan(x/2)=sinx/(1+cos x)

QED

Answer 2

prove (sin2x-sinx)/(cos2x+cosx)=sinx/(1+cosx)

sin2x=2sinxcosx (trig identity)

cos2x=2cos^2x-1 (trig identity)

substitute

(2sinxcosx-sinx)/(2cos^2x-1+cosx)=

factor both terms

sinx(2cosx-1)/(2cosx-1)(cosx+1)=

simplify the fraction-cancel common terms

sinx/(cosx+1) which is the right-hand side