Expert: Paul Klarreich - 10/12/2007

Question
sinxcos^3x-cosxsin^3x=1/4sin4x

Verify the identity.

Thanks a million! These identities are killing me.

Answer
Questioner:   Cherry
Category:  Calculus
Private:  No
 
Subject:  trig identities
Question:  sinxcos^3x-cosxsin^3x=1/4sin4x

Verify the identity.

Thanks a million! These identities are killing me.
............................................
Hi, Cherry,

I think you mean:

sin x cos^3(x) - cos x sin^3(x) = sin4x / 4

Here you need your basic double-angle formulas:

sin(2x) = 2 sin x cos x
cos(2x) = cos^2(x) - sin^2(x), among other things.

Let's go:

sin(4x)
------- =
 4

2 sin(2x) cos(2x)
------------------  << apply a double-angle formula
       4

2 (2 sin x cos x)(cos^2(x) - sin^2(x))
-------------------------------------- << and again.
          4

4 sin x cos x (cos^2(x) - sin^2(x))
----------------------------------- << multiply out
          4

sin x cos x (cos^2(x) - sin^2(x))  << cancel the 4

sin x cos^3(x) - cos x sin^3(x)   << multiply out.