<s>[INST] Expand using binomial theorem Yesterday, my maths lecturer did this:
$[(\cos x+i\sin x)+(\cos x-i\sin x)]^4$
Expanded to:
$8\cos2x+2\cos4x+6$
I've tried using binomial theorem to get to this answer but I can never seem to get there. How can I do it?
When I do it I get to:
$2\cos4x+4(\cos3x+i\sin3x)(\cos x-i\sin x)+6(\cos^22x+\sin^2(2x))+4(\cos^2x-i^2\sin^2x)$
 [/INST] If we combine like terms, we get $(2\cos x)^4$.  The Double-angle formula for $\cos$ is: $$\cos 2x=2\cos^2x-1$$  We start with $(2\cos x)^4=16\cos^4x$ and get $4(2\cos^2x)^2=4(\cos2x+1)^2=4\cos^22x+8\cos 2x+4,$ and the first term gives $2(2\cos^22x)=2\cos4x+2$, for a grand total of $2\cos 4x+8\cos 2x+6$.
From where you left off, it is possible to use $\cos^2 a+\sin^2 a=1$ to reduce a good portion of your binomial results to a simple value of $10$, after which you can focus on the not-yet-distributed portion of your results.
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