Prove $\sin^2\theta + \cos^2\theta = 1$ How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$
I'm curious to know of the different ways of proving this depending on different characterizations of sine and cosine.

In the spirit of Git Gud's answer, differentiate $\sin^2 \theta + \cos^2 \theta$ to get
$$ 2 \sin \theta \cos \theta - 2 \cos \theta \sin \theta = 0$$
So $\sin^2 \theta + \cos^2 \theta$ is constant. Plugging in $\theta = 0$ shows that constant is $1$.
