Question

How do you prove that (sin 5x)^2 - (sin3x)^2 = sin8x sin2x?

Answer 1

f(x) = (sin 5x - sin 3x)(sin 5x + sin 3x) .
Use trig identities:
`sin a - sin b = 2cos ((a + b)/2)sin ((a - b)/2)`
`sin a + sin b = 2sin ((a + b)/2)cos ((a - b)/2)`
In this case:
sin 5x - sin 3x = 2cos 4x.sin x
sin 5x + sin 3x = 2sin 4x.cos x.
We get:
f(x) = 2sin 4x.cos 4x.(2sin x.cos x) = sin 8x.sin 2x
Reminder:
cos 8x = 2sin 4x.cos 4x
sin 2x = 2sin x.cos x