<s>[INST] Derive identities for $\cos(4x)$ and $\sin(4x)$ using following fact So I need to use the fact that: $$\cos(4x) + i\sin(4x) = \left(\cos(x) + i\sin(x)\right)^4$$ to derive identities for $\cos(4x)$ and $\sin(4x)$ in terms of $\cos(x)$ and $\sin(x)$. I'm not sure how to go about this, could I please get some help.
 [/INST] $$(\cos x +i \sin x)^4=\cos 4x+i \sin 4x$$
$$(\cos x +i \sin x)^4=\cos^4x+4\cos ^3x \cdot (i \sin x)+6\cos ^2x \cdot (i \sin x)^2+4\cos x \cdot (i \sin x)^3+(i \sin x)^4=$$
$$=\cos^4x-6 \cos ^2x \sin^2x+\sin^4x+$$
$$+i(4 \cos ^3 x \sin x-4\cos x\sin^3x)$$
Then $$\cos 4x=\cos^4x-6 \cos ^2x \sin^2x+\sin^4x$$
and $$\sin 4x=4 \cos ^3 x \sin x-4\cos x\sin^3x$$
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