Question

prove that sin6x+sin2x+2sin4x/sin7x+sin3x+2sin5x=sin4x/sin5x​

Answer 1

Answer:

Keep using the identity sin(A + B) = sinAcosB + cosAsinB and sin(A - B) = sinAcosB - cosAsinB.

sin(x) + sin(3x) + sin(5x) + sin(7x)

= sin(2x-x) + sin(2x+x) + sin(6x-x) + sin(6x+x)

= sin(2x)cos(x) - cos(2x)sin(x) + sin(2x)cos(x) + cos(2x)sin(x) + sin(6x)cos(x) - cos(6x)sin(x) + sin(6x)cos(x) + cos(6x)sin(x)

= 2cosx[sin(2x) + sin(6x)]

=2cosx[sin(4x-2x) + sin(4x+2x)]

=2cosx[sin(4x)cos(2x) - cos(4x)sin(2x) + sin(4x)cos(2x) + sin(4x)cos(2x)]

=4cos(x)cos(2x)sin(4x)