Question

How do you prove #(sinx-2sin2x+sin3x)/(sinx+2 sin2x+sin3x)= -tan^2(x/2)#?

Answer 1

#LHS=(sinx-2sin2x+sin3x)/(sinx+2 sin2x+sin3x)#

#=(2sin2xcosx-2sin2x)/(2sin2xcosx+2 sin2x)#

`=-(2sin2x(1-cosx))/(2sin2x(1+cosx))`

`=-(2sin^2(x/2))/(2cos^2(x/2))`

`= -tan^2(x/2)=RHS`