# Proving a Trigonometric Identity

## Homework Statement

Prove that:

$\cos^6{(x)} + \sin^6{(x)} = \frac{5}{8} + \frac{3}{8} \cos{(4x)}$

## Homework Equations

I am not sure. I used factoring a sum of cubes.

## The Attempt at a Solution

I tried $\cos^6{(x)} + \sin^6{(x)} = \cos^4{(x)} - \cos^2{(x)} \sin^2{(x)} + \sin^4{(x)}$. But I can't get anywhere beyond this; I must be missing something obvious.

HallsofIvy
Homework Helper
Sounds good to me! Now, you might try factoring "$cos^2(x)$ out of the first two terms: $cos^2(x)(cos^2(x)- sin^2(x))+ sin^4(x)= cos^2(x)cos(2x)- sin^4(x)$ see where you can go from that.

Simon Bridge
Homework Helper
Id normally just throw the euler formula at these things ... unless I had an already proved identity I could use.

SammyS
Staff Emeritus
Homework Helper
Gold Member
Although, x2 - xy + y2 cannot be factored (over the reals), x4 - x2 y2 + y4 can be factored .

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SteamKing
Staff Emeritus
Homework Helper
Rather than attacking the LHS of the identity, I would prefer to look at the expression cos (4x) instead. The multiple angle formulas for cosine I think would be more helpful here than trying to factor polynomials.

haruspex