Prove the identity involving $cos( heta)$ and $sin( heta)$ on the unit circle

To prove the identity involving $ \cos(\theta) $ and $ \sin(\theta) $ on the unit circle, we start with the Pythagorean identity:

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

Consider the parameterization of the unit circle with $ \theta $ as the angle from the positive x-axis:

$ x = \cos(\theta) $

$ y = \sin(\theta) $

Then, the coordinates $ (x, y) $ must satisfy:

$ x^2 + y^2 = 1 $

Substituting $ x = \cos(\theta) $ and $ y = \sin(\theta) $, we get:

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

This verifies the identity.

To prove the identity involving $ cos( heta) $ and $ sin( heta) $ on the unit circle, we use the Pythagorean theorem:

$ cos^2( heta) + sin^2( heta) = 1 $

Given that on the unit circle, $ x = cos( heta) $ and $ y = sin( heta) $, the coordinates must satisfy:

$ x^2 + y^2 = 1 $

Substitute $ x $ and $ y $:

$ cos^2( heta) + sin^2( heta) = 1 $

On the unit circle, it holds that:

$ cos^2( heta) + sin^2( heta) = 1 $

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