Use the trig identities to transform the right side:
sin 2x = sin x.cos x
cos 2x = 2cos^2 x - 1cos2x=2cos2x−1.
Transform the right side:
RS = (sin 2x + sin x)/(cos 2x - cos x) = (sin x(2cos x + 1))/(2cos^2 x - cos x - 1sin2x+sinxcos2x−cosx=sinx(2cosx+1)2cos2x−cosx−1 =
Factor the trinomial (2cos^2 x - cos x - 1).(2cos2x−cosx−1).
Since (a + b + c = 0), use the Shortcut, the 2 factors are (cos x - 1) and (2cos x + 1). Finally,
RS = (sin x(2cos x + 1))/((cos x - 1)(2cos x + 1)) = (sin x)/(cos x - 1)RS=sinx(2cosx+1)(cosx−1)(2cosx+1)=sinxcosx−1
Since:
sin x = 2sin (x/2).cos (x/2) andsinx=2sin(x2).cos(x2)and
(cos x - 1) = -2sin^2 (x/2)(cosx−1)=−2sin2(x2), therefor
RS = (2sin (x/2)(cos x/2))/(-2sin^2 (x/2)) = RS=2sin(x2)(cosx2)−2sin2(x2)=
= - cos (x/2)/(sin (x/2)) = -cot x/2=−cos(x2)sin(x2)=−cotx2
