On the unit circle, is the x-coordinate and is the y-coordinate of the point at angle . Add them directly and the sum swings positive and negative as the point moves around the circle. Square them first, and the picture changes completely.

Concept A sum vs. a sum of squares

can be positive or negative — it depends on where sits. But squaring removes the sign: any real number squared is never negative, so both and always sit between 0 and 1. Their sum is fixed:

Concept Why it's Pythagoras

The radius to any point on the unit circle has length 1.

That radius is the hypotenuse of a right triangle with legs

and

The Pythagorean theorem gives

directly.

Example Checking it at the axes

At θ = 0°

⟹ 1² + 0² = 1

At θ = 90°

⟹ 0² + 1² = 1

At either axis, one term does all the work and the other vanishes — but the total never changes.

Note The sum wanders, the sum of squares doesn't

θ	sin θ + cos θ	sin²θ + cos²θ
0°	1	1
180°	−1	1
225°	≈ −1.41	1

The plain sum changes value and sign as sweeps around the circle. The sum of squares never moves — it is an identity, true for every angle.

Summary

changes continuously and can be positive or negative.
Squaring removes the sign, so and always lie between 0 and 1.
The Pythagorean identity: for every angle .
It is Pythagoras applied to the unit circle: legs and , hypotenuse 1.
At the axes one term carries the whole total; elsewhere the two squared terms share it, always summing to 1.

The Pythagorean Identity