<s>[INST] In how many different ways can you prove that $\sin^2x + \cos^2x = 1$ The standard proof of the identity $\sin^2x + \cos^2x = 1$ (the one that is taught in schools) is as follows: from pythagoras theorem, we have (where $h$ is hypotenuse, $b$ is base and $p$ is perpendicular)
$$h^2 = p^2 + b^2$$
dividing by $h^2$ on both sides:
$$1 = \frac{p^2}{h^2}+\frac{b^2}{h^2}$$
since $\sin x = \frac ph$ and $\cos x = \frac bh$, 
$$1 = \sin^2x+\cos^2x$$
Are there any more innovative ways of proving this common identity?
 [/INST] A few proofs I came up with are:
Proof 1: This uses real analysis. Let $f(x) = \sin^2x + \cos^2x \implies f'(x) = 2\sin x \cos x - 2\sin x \cos x = 0$. Thus, $f(x)$ is a constant function. To find the value of $f(x)$, it is sufficient to find $f(c)$, where c is any convenient constant (say 0). Therefore, $f(x) = f(0) = 1$
Proof 2: This uses Euler's representation of complex numbers. We can represent a complex number in the form $z = re^{ix}$, where $r$ is the modulus and $z$ is the argument. If we take the number $z = e^{ix} = \cos x + i\sin x$ (De Moivre's theorem), then $|z| = 1 = \sqrt{\cos^2x + \sin^2x} \implies \cos^2x + \sin^2x=1$.
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