## Prove the identity sin(4x) = 8sin(x)cos^3(x)-4sin(x)cos(x)

Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.

### Prove the identity sin(4x) = 8sin(x)cos^3(x)-4sin(x)cos(x)

This is another problem which I am not able to complete

Prove the following identity.

sin(4x) = 8sin(x)cos^3(x)-4sin(x)cos(x)
MathRaven

### Re: Prove the identity sin(4x) = 8sin(x)cos^3(x)-4sin(x)cos(

MathRaven wrote:This is another problem which I am not able to complete

how did you start it?

MathRaven wrote:Prove the following identity.

sin(4x) = 8sin(x)cos^3(x)-4sin(x)cos(x)

how i did it: factor out from the RHS & use one double-angle ident twice.
buddy

Posts: 53
Joined: Sun Feb 22, 2009 10:05 pm

### Re: Prove the identity sin(4x) = 8sin(x)cos^3(x)-4sin(x)cos(

MathRaven,

It helps if you have a list of identities to choose from. I started with
sin 4x = 2 sin 3x cos x - sin 2x and then used sin 3x = (sin x) (-1 + 4cos^2x), so using this substitution we get
= 2 [(sin x) (-1 + 4cos^2x)]cos x - sin 2x
= 2[sinx cos x (-1 + 4cos^2x)] - sin 2x
= 2 [ -sinx cosx + 4 sinx cos^3x] - sin 2x
= -2sinx cosx + 8 sin x cos^3x - sin 2x and now using sin 2x = 2 sinx cosx give us
= - 2sinx cosx + 8 sinx cos^3x - 2sinx cosx
= 8 sinx cos^3x - 4 sinx cosx

I think identities are fun, don't you?
LoveMath

Posts: 7
Joined: Mon Sep 03, 2012 4:35 am