Deriving the Pythagorean Identities

Main Identity :                                       cos^2 (-) + sin^2 (-) = 1

Back to Unit Circle:

So in the Pythagorean theorem the square of the long side equals the square of the two other sides added together.
                             ex    x^2+y^2=1
On the other hand the unit circle, the longest side is the radius which always equals 1. X is our cosine while Y is our sine. so they added together they will equal 1.
                                     ex   cos^2 (-) + sin^2 (-) = 1

Deriving the Two Remaining Pythagorean Identities:
The pictures above will show you how.