Prove the trigonometric identity $\sin^4{x} = \frac{3-4\cos{2x}+\cos{4x}}{8}$

Question

I need to show the steps to prove this identity: $$\sin^4{x} = \frac{3-4\cos{2x}+\cos{4x}}{8}$$

I know that $\cos{2x}=\cos^2{x}-\sin^2{x}$. From there I do not know what to do. The solution should look like:

$$\sin^4{x}=sin^4{x}$$

I need to prove the right side equals the left side.

Answer 1

If you know $\sin x = \frac{e^{ix}-e^{-ix}}{2i}$ and $\cos x = \frac{e^{ix}+e^{-ix}}{2}$, then just expand using the binomial theorem: $$\sin^4 x = \frac{e^{4ix} - 4e^{3ix}e^{-ix}+6e^{2ix}e^{-2ix}-4e^{ix}e^{-3ix}+e^{-4ix}}{(2i)^4}$$ $$=\frac{e^{4ix}+e^{-4ix} -4e^{2ix}-4e^{-2ix} + 6}{16}$$ $$=\frac{\frac{e^{4ix}+e^{-4ix}}{2} -4\frac{e^{2ix}+e^{-2ix}}{2} + 3}{8}$$ $$=\boxed{\frac{\cos 4x -4\cos 2x + 3}{8}}$$ as desired.

Answer 2

Hint: $$\cos{2x}=\cos^2{x}-\sin^2{x}=1-2\sin^2{x}\\ \implies \sin^2{x}=\frac{1-\cos{2x}}{2}$$

Answer 3

Hint: $(\cos^2 x-\cos2x)(\cos^2 x-\cos 2x)=(\sin^2 x)^2$

Answer 4

Let $ z = \cos(\theta) +i \sin(\theta)$

We can then find through De Moive's formula and the fact that $\cos -\theta = \cos \theta$ and $\sin -\theta = - \sin \theta$:

$ z - \frac1z \equiv 2i\sin\theta$

Thus: ($i^4=1$)

$ (z-\frac1z)^4 \equiv 16\sin^4\theta$

Expanding the RHS:

$ (z-\frac1z)^4 \equiv z^4 - 4z^2 + 6 - 4z^{-2} + z^{-4}$

Then subsituting $z$ back into the equation. (The $\sin$s will cancel out)

$ (z-\frac1z)^4 \equiv 2\cos4\theta - 8\cos2\theta + 6$

Finally dividing by 16 to get $\sin^4\theta$:

$\sin^4\theta = \frac {\cos4\theta - 4\cos2\theta + 3} 8$

Answer 5

You could also use the fact that $\text{Re} ((\cos x + i \sin x)^4) = \text{Re} (\cos 4x + i \sin 4x)$.