How to prove the identity `sin^2x + cos^2x = 1` ?

This is not a proof, but it sure is compelling evidence:

Enter y=(sin(x))^2+(cos(x))^2 into a graphing calculator and look at the result for -2pi < x < 2pi.  Yep, it's the constant function y=1.

Or think about it without technology.  What would the graph of f(x)=(sin(x))^2 look like?  All points with y-values of 0 or 1 would not change, and points with y=-1 would keep their x-values but get y-values of +1.  So this graph would be zero at even multiples of pi/2 and 1 at odd multiples of pi/2.  A similar analysis of g(x)=(cos(x))^2 gives a graph that is 1 where f(x) is 0, and 0 where f(x) is 1.  So y=f(x)+g(x) is clearly 1 in all of those places.

Again, it's not a proof, but it's good to have a graphical look at this important trig identity.