Math Question-Please help!?

Question

Verify (2cos3x)(sinx)= (2sinx)(cosx)-(8cosx)(sin^3x)

I've been working on this question forever and can't seem to figure it out. Would appreciate any help.

Answer 1

First, I note that each term is multiplied by a sin(x), so I will just work with getting 2cos(3x) to become 2cos(x) - 8cos(x)sin²(x)

Using the identity cos(a+b) = cos(a)cos(b) - sin(a)sin(b) with a=2x and b=x,

2cos(3x) = 2cos(2x+x) = 2[cos(2x)cos(x) - sin(2x)sin(x)]

Using the identities cos(2x) = 1 - 2sin²(x) and sin(2x) = 2sin(x)cos(x),

2cos(3x) = 2[1 - 2sin²(x)]cos(x) - 4sin²(x)cos(x)

Expanding and combining like terms,

2cos(3x) = 2cos(x) - 4sin²(x)cos(x) - 4sin²(x)cos(x) = 2cos(x) - 8sin²(x)cos(x)

Now multiply both sides by sin(x) and you get the original identity.