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.