How do you derive cos^2(x) plus sin^2(x)=1?

Think of how the trigonometric functions are defined for a right-angled triangle.

Cos(theta) = adjacent/hypotenuse.
sin(theta) = opposite/hypotenuse.

sin^2 + cos^2 = o^2/h^2 + a^2/h^2
= (o^2 + a^2)/h^2

But, from the pythagorean theorem,

o^2 + a^2 = h^2

Therefore,

cos^2 + sin^2 = (o^2 + a^2)/h^2 = 1