Prove; sinx+sin3x+sin5x= sin^2(3x)/sinx?

Question

Please help me prove this identity. I really need the step by step solution of this problem. thank you!

Answer 1

sin3x/sinx
= sin(x+2x)/sinx
= (sinxcos2x + cosxsin2x)/sinx
= cos2x+2cos^2(x)
= cos2x+cos2x+1
= 2cos2x+1
=> 2cos2x+1 = sin3x/sinx

sinx+sin3x+sin5x
= sin(3x-2x)+sin3x+sin(3x+2x)
= 2sin3xcos2x+sin3x
= sin3x(2cos2x+1)
= sin^2(3x)/sinx