trigonometry - Confirmation of Proof: $cos^2(x) = frac 12big(1+cos(2x)big)$ using Euler's identity?

How can I prove the following equation:

$$\cos^2(x) = \frac 12\big(1+\cos(2x)\big),\tag1$$ using Euler's identity? $$e^{i\pi} + 1 = 0.\tag*{$\begin{align} \because e^{i\theta} &= \cos\theta + i\sin\theta \\ &= \text{cis} \ \theta \end{align}$}$$

I have tried equating Euler's equation to cos on one side and squaring that but haven't had luck reducing it to the desired form as outlined in $(1)$.