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Let's start with the Unit Circle:

unit_circle.gif

Zooming in to get a better look, we can label the sides of our right triangle...

unit_circle2.gif

By the Pythagorean Theorem, we have that

pythagorean_thm.gif

Since this is a Unit Circle, we have

pythagorean_equals.gif         unit_circle3.gif

Substituting, gives us

pythagorean_thm2.gif

pythagorean_identity1.gif

pythagorean_identity1a.gif
(The first Pythagorean Identity)