Sum of infinite series involving cos$theta$

Find the sum of following series:

$$1 + \frac{1}{4!} \cos 4\theta + \frac{1}{8!} \cos 8\theta + ...$$

My attempt:

I need hint to start. Thanks!

Answer

HINT
$$\frac 12(\cosh(x)+\cos(x))=\frac 12\left(\sum_{k=0}^\infty \frac {x^{2k}}{(2k)!}+\sum_{k=0}^\infty \frac {(-1)^kx^{2k}}{(2k)!}\right)=\sum_{k=0}^\infty \frac {x^{4k}}{(4k)!}$$
and you have:
$$\sum_{k=0}^\infty \frac{\cos(4k\theta)}{(4k)!}$$
and $\cos(x)=\Re(e^{ix})$

The answer turns out to be: $\dfrac 12 \left(\cos(\sin(\theta))\cosh(\cos(\theta))+ \cos(\cos(\theta))\cosh(\sin(\theta))\right)$