Prove the Identity (step by step)

Question

sin(x) + sin(3x) = 4 sin(x) cos^2(x)

Answer 1

Here is another way to look at it--transform left side into right side.

 

sinx + sin (3x)

 

sinx + sin (2x +x)

 

sinx + sin2xcosx + cos2x sinx   (use sin(A+B) identity)

 

sinx + 2sinxcosxcosx + (2cos²x-1)sinx    {sin2x identity and cos2x identity)

 

sinx + 2sinxcos²x + 2sinxcos²x - sinx 

 

4 sinx cos²x 

 

Q.E.D.

 

 

Answer 2

The formula to use is sinA+sinB=2sin((A+B)/2)cos((A-B)/2)

sin(x) + sin(3x) = 4 sin(x) cos^2(x)

2sin((x+3x)/2)cos((3x-x)/2)=2sin2xcosx=4sinxcosxcosx=4sinxcos²x