Proof of $\sin^2 x+\cos^2 x=1$ using Euler's Formula How would you prove $\sin^2x + \cos^2x = 1$ using Euler's formula?
$$e^{ix} = \cos(x) + i\sin(x)$$
This is what I have so far:
$$\sin(x) = \frac{1}{2i}(e^{ix}-e^{-ix})$$
$$\cos(x) = \frac{1}{2} (e^{ix}+e^{-ix})$$

you just need to finish it.
$\sin^2(x) = \frac{1}{-4}  ((e^{ix})^2 - 2 e^{ix}e^{-ix} + (e^{-ix})^2)$
$\cos^2(x) =\frac{1}{4}  ((e^{ix})^2 + 2 e^{ix}e^{-ix} + (e^{-ix})^2)$
$\sin^2(x) + \cos^2(x) = \frac{1}{4}  4 e^{ix}e^{-ix} = 1$
