I'm trying to create a proper explanation in the cartesian coordinate system as to why $\sin^2 x + \cos^2 x = 1$.
Although I could go with the derivative, I figured that it'd be a nice alternative if I could show that the two functions are symmetrical to $0.5$, but cannot for the life of me proceed from there
Any recommendations?
One visual interpretation could be plotting the unit circle centered at origin in the Cartesian plane and then switching to polar coordinates.
Let $P(x,y)$ be a point on the circle. Then, $x^2+y^2=1$.
Switching to polar coordinates, $x=\cos\theta$ and $y=\sin\theta$ with $0\leq\theta\lt 2\pi$ which shows $\sin^2\theta+\cos^2\theta=1~\forall~\theta\in [0,2\pi)$.
Now, since $2\pi$ is a period for any circular trigonometric functions, this shows the above holds for all $\theta\in\Bbb R$ (if $\theta\in [0,2\pi)^c$, think of the angle mod $2\pi$ where you take the non-negative remainder as $\theta$; the angle $\theta$ is increasing counter-clockwise).