How to Prove that sin^2(x) + cos^2(x) = 1

First, we need to consider a unit circle with a radius of 1 and a center of (0,0) on the x-y plane. Note that the angle shown uses theta instead of x. 

In this diagram, every point on the circle can be in the form of the cosine of the angle theta for the x coordinate and sine of the angle theta for the y coordinate. 

For the point shown, we can draw a right triangle using it and the origin as two of the corners. The length of the hypotenuse is 1, and the lengths of the other two sides are cosine of theta and sine of theta.

Using the Pythagorean theorem (a^2 + b^2 = c^2), we can say that the square of the sine of theta plus the square of the cosine of theta must equal the square of the hypotenuse which is 1. Putting that all together, we get:

sin^2(x) + cos^2(x) = 1^2

which simplifies to:

sin^2(x) + cos^2(x) = 1

QED

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Q.E.D. or QED is an initialism of the Latin phrase quod erat demonstrandum, meaning "which was to be demonstrated". Literally it states "what was to be shown".[1] Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in print publications, to indicate that the proof or the argument is complete.

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