I was asked to find a formula for $\cos(5x)$ in terms of $\sin(x)$ and $\cos(x)$.
I tried to use the formula $\cos(5x) + i\sin(5x) = (\cos(x)+i\sin(x))^5
$ and what I get is
$16i\sin^5(x) – 20i\sin^3(x) + 5i\sin(x) + \cos(x) + 16 \sin^4(x) \cos(x) – 12 \sin^2(x) \cos(x)$
But how do I deal with the $i\sin(5x)$? Because I only can use $\sin(x)$ and $\cos(x)$ to express $\cos(5x)$. Thank you for your help!
Best Answer
Your way is right. But I think it should be
$$(\cos x + i\sin x)^5 = \cos^5 x + 5i\cos^4x\sin x - 10\cos^3x\sin^2 x - 10i\cos^2x\sin^3x + 5\cos x\sin^4x +i\sin^5x$$
So, you have $$\cos 5x = \cos^5x-10\cos^3x\sin^2x+5\cos x\sin^4x$$ $$\sin 5x = \sin^5x-10\cos^2x\sin^3x+5\cos^4 x\sin x$$