Use De Moivre's Theorem to express $\cos(4θ)$ in terms of sums and differences of powers of $\sin(θ)$ and $\cos(θ)$ So far I have gotten this far using the binomial theorem. 
$(x + iy)^4 = x^4 + 4x^3y + 6i^2x^2y^2 + 4i^3xy^3 + i^4y^4$=
$x^4 - 6x^2y^2 + y^4 + i(4x^3y - 4xy^3)$
$i^2 = -1 $ let $x = cosθ$  and y = $sinθ$
$e^{i4θ} = cos4θ + isin4θ = (cosθ + i sinθ)^4 $
$cos4θ = cos^4θ - 6cos^2θsin^2θ + sin^4θ$
I am not quite sure where to go next from here? Have I expressed this correctly in terms of sinθ and cosθ? 
Thanks for taking the time to review. 
 A: Your calculations are correct, but there are a few notational details. Firstly, a few misprints, for instance, $e^i40 = \cos 4x + i\sin x$ is probably meant to be $(e^i)^4 = \cos 4x + i\sin 4x$, or something like it.
Secondly, you're using the letter $x$ to do two diffferent things here. I would probably let the first line be $(a + ib)^4$ instead, and then insert $a = \cos x, b = \sin x$ afterwards.
Lastly, there is a separate command to make the sine and cosine look nice within a math environment (between dollar signs \$ \$). So instead of writing sin x or sin(x), which gives $sin x$ or $sin(x)$ respectively, you write \sin x or \sin (x), which renders like this: $\sin x$, $\sin (x)$. Similarily, you can write \cos x to get $\cos x$, and there are a lot more trigonometric functions like this (\tan x, \arcsin y, \cosh z), not to mention other common mathematical terms, like logarithms (\ln and \log), max and min and similar functions (\max, \min, \sup, \inf) and so on. If you want to make your own (if what you want doesn't exist already) you use the following command: \operatorname{Func}x. It renders like this: $\operatorname{Func} x$.
A: ${(\cos{\theta}+i\sin{\theta})^4}\to$
${(c+is)^4}\to$  (for brevity)
$\begin{matrix}
1&4&6&4&1\\
c^4&c^3&c^2&c&1\\
1&is&-s^2&-is^3&s^4\\
\hline
c^4&4ic^3s&-6c^2s^2&-4ics^3&s^4
\end{matrix}$
The terms pertaining to $\cos(4\theta)$  are the real ones; the imaginaries, after dispensing with $i$, pertain to $\sin(4\theta)$ .
So, $\cos(4\theta)=\cos^4\theta-6\cos^2\theta\sin^2\theta+\sin^4\theta$ .
And, taking advantage of the fundamental relation between sine and cosine, $\cos(4\theta)=8\cos^4-8\cos^2+1=8\sin^4\theta-8\sin^2\theta+1$
