Why is the name Pythagorean Identity appropriate?

When you think of Pythagorean Identity, the first thing that pops into your head is probably the Pythagorean Theorem. If so, then that's great! This is exactly where the term Pythagorean Identity came from!

 Now that we have a triangle with  a hypotenuse of 1, we can use the Pythagorean theorem to derive the special trigonometry functions that we will use to form the Pythagorean Identity.

So here's our formula: a^2+b^2=c^2
(for our purposes, a will be x and b will be y)
Now let's find the trig functions. We know the following:
Sin()= opposite / hypotenuse = y / 1 = y
Cos()= adjacent / hypotenuse = x / 1 = x

By substituting this into our triangle, our result will be the following:

And now that we have our trigonometry functions, all we have to do is plug them in to the Pythagorean Theorem! 

Which would look like the following: 

a^2+b^2=c^2  (remember for our purposes, for our previous triangle: a=x and b=y) 

cos^2+sin^2=1 --> which is the first Pythagorean Identity!

So I hope you now understand how the Pythagorean Identity got its name, because of the Pythagorean Theorem!