# Sum of Squares of Sine and Cosine/Proof 3

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## Theorem

- $\cos^2 x + \sin^2 x = 1$

## Proof

Let $P = \left({x, y}\right)$ be a point on the circumference of a unit circle whose center is at the origin of a cartesian coordinate plane.

From Sine of Angle in Cartesian Plane and Cosine of Angle in Cartesian Plane:

- $P = \left({\cos \theta, \sin \theta}\right)$

The graph of the unit circle is the locus of:

- $x^2 + y^2 = 1$

as given by Equation of Circle.

Substituting $x = \cos \theta$ and $y = \sin \theta$ yields:

- $\cos^2 \theta + \sin^2 \theta = 1$

$\blacksquare$

## Notes

This proof requires that we start from the unit circle definitions of sine and cosine, otherwise the proof is circular.