Sum of Squares of Sine and Cosine/Proof 3

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


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

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

$P = \tuple {\cos \theta, \sin \theta}$

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:

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



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