8. Trigonometric identities

Euler's formula can be used as a shortcut.

There are several ways to come up with an expression for $\cos(4x)$. Euler's formula provides a straightforward way. Let's recall $$ e^{ix} = \cos(x) + i\sin(x). $$

As a result we can write $$ \cos(4x) = \mathfrak{Re}(e^{i4x}). $$

Furthermore, we have $$ e^{i4x} = (e^{ix})^4 = (\cos(x) + i\sin(x))^4. $$

So we just need to evaluate the binomial expansion above, that is $$ (\cos(x) + i\sin(x))^4 = \cos^4(x) + 3i\cos^3(x)\sin(x) $$ $$ -6\cos^2(x)\sin^2(x) - 3i\cos(x)\sin^3(x) + \sin^4(x). $$

Finally, taking the real part of the above expression results in $$ \cos(4x) = \cos^4(x) -6\cos^2(x)\sin^2(x) + \sin^4(x). $$

The sum of the coefficients is $1-6+1=-4$.