Prove that (sinx)^2 + (cosx)^2 = 1

We start with the definitions of sine and cosine, which are, respectively: sinx = opposite/hypoteneuse and cosx = adjacent/hypoteneuse. We then square the analyzed expressions to get the following: 

(opposite ^2)/(hypoteneuse ^2) + (adjacent ^2)/(hypoteneuse ^2)

And since the denominators are the same, we can add the fractions to get: 

(opposite ^2) + (adjacent ^2) / (hypoteneuse ^2)

But recall the Pythagorean Theorem, according to which: (opposite ^2) + (adjacent ^2) = (hypoteneuse ^2). So we get:

[(hypoteneuse ^2)] / (hypoteneuse ^2) = 1. QED.