Primary Pythagorean Identity

(1)

To verify this identity, we start by rewriting it

(2)

the diagram

which is mathematically identical. But now, the equation takes the form of the Pythagorean theorem, where a right triangle, whose hypotenuse's length is 1, has legs with lengths of a and o, as shown in the diagram.

In terms of the diagram,

(3)

and

(4)

Substituting with these definitions, eqn. 2 becomes

(5)

in this case h = 1, so the left side may be simplified to

(6)

and, for the same reason, h can be introduced on the right in place of 1, yielding

(7)

Rearranging and renaming, where leg lengths are now denoted a and b, and the hypotenuse length is now c, gives the standard form of the Pythagorean theorem.

(8)

So, eqn. 1 is, therefore, a manifestation of this theorem in the case where c = 1. This means that based on the Pythagorean theorem,

(1)