trigonometry - Help with using Euler’s formula to prove that $cos^2(theta) = frac{cos(2theta)+1}{2}$

I have to use Euler's Formula to prove that:

$$\cos^2(\theta) = \frac{\cos(2\theta)+1}{2}.$$

I have managed to prove this using trigonometric identities but I'm not sure how to use Euler's Formula or how it links into the question.

My method so far has been:

$$\frac{(\cos(2\theta)+1)}{2} = \frac{(\cos^2(\theta) - \sin^2(\theta)+1)}{2}$$

since

$$\cos(2\theta)=\cos(\theta)\cos(\theta)-\sin(\theta)\sin(\theta).$$

So
$$\frac{(\cos(2\theta)+1)}{2} =\frac{2\cos^2(\theta)}{2}
=\cos^2(\theta).$$

Answer

Eulers identity $e^{i\theta} = \cos \theta + i\sin\theta$

$e^{i\theta} + e^{-i\theta} = 2\cos \theta\\
\frac 14 (e^{i\theta} + e^{-i\theta})^2 = \cos^2 \theta\\

\frac 14 (e^{2i\theta} + e^{-2i\theta} + 2) = \cos^2 \theta\\
\frac 14 (2\cos 2\theta + 2) = \cos^2 \theta\\
\frac 12 (\cos 2\theta + 1) = \cos^2 \theta$