[Math] Using Complex exponential definitions of sine and cosine, prove $\cos2\theta=\cos^2\theta-\sin^2\theta$

Question

Question

Please do not just tell me the answer, please provide helpful hints and hide the answers

Using Complex exponential definitions of sine and cosine, prove $\cos\theta=\cos^2 \theta-\sin^2\theta$

All that I know is the trig identity:

$\cos2\theta=1-2\sin^2\theta$ or $\cos2\theta=2\cos^2\theta-1$

Answer 1

Hint:

$\cos 2\theta + i \sin 2\theta = e^{2i\theta} =\left(e^{i\theta}\right)^2=(\cos \theta+ i\sin \theta)^2$

Then expand the right side and compare real parts