Sum of Squares of Sine and Cosine/Proof 3
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Theorem
- $\cos^2 x + \sin^2 x = 1$
Proof
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:
- $\cos^2 \theta + \sin^2 \theta = 1$
$\blacksquare$
Notes
This proof requires that we start from the unit circle definitions of sine and cosine, otherwise the proof is circular.