Why is it that sin^2(x) + cos^2(x) = 1?

Let's start with the definition of sin(x) and cos(x). We're going to use t rather than x here to avoid confusion, but the idea is exactly the same.

Take a unit circle - that is, a circle with centre (0,0) and radius 1 - and pick any point on it. Let the co-ordinates of this point be (x,y). Now draw a line from the origin to your point. Start from the point, and move the (real or imaginary) pencil round clockwise until you reach (1,0) on the positive x-axis. What you've just drawn is t, which is the angle between your line and the positive x-axis.

By definition, x = cos(t) and y = sin(t). So the co-ordinates of your point could equally be written (cos(t), sin(t)).

Now we can form a right-angled triangle with side lengths x, y and 1. It's clear from this that x² + y² = 1, or in other words, cos²(t) + sin²(t) = 1 for all values of t. Here is a diagram to clarify: goo.gl/YuLWqe

You sometimes see this written as cos²(t) + sin²(t) ≡ 1. The "triple bar" equals sign just emphasises that it is true for all values of t.