Using Pythagoras' theorem, show that sin^2(x)+cos^2(x)=1 for all x.

Take a right angled triangle with hypotenuse of length 1, and angle at the bottom of the hypotenuse equal to x. We will let o denote the length of the side opposite the angle, and a denote the length of the side adjacent to the angle.
Using SOHCAHTOA, we know that sin(x)=o/1=o, and cos(x)=a/1=a.
So we now have a right angled triangle with a hypotenuse of length 1, another side of length sin(x), and a side of length cos(x). Using Pythagoras' theorem, we know that o^2+a^2=1^2, and so sin^2(x)+cos^2(x)=1.