Determine the coordinates of points on the unit circle that satisfy the equation $ cos^2( heta) - sin^2( heta) = 0 $

To determine the coordinates of points on the unit circle that satisfy the equation $ \cos^2(\theta) – \sin^2(\theta) = 0 $:

First, we recall the Pythagorean identity: $ \cos^2(\theta) + \sin^2(\theta) = 1 $

Given the equation: $ \cos^2(\theta) – \sin^2(\theta) = 0 $

This can be rewritten as: $ \cos^2(\theta) = \sin^2(\theta) $

Taking the square root of both sides gives: $ \cos(\theta) = \pm \sin(\theta) $

We consider the positive and negative cases separately.

For $ \cos(\theta) = \sin(\theta) $:

$ \theta = \frac{\pi}{4} + k \pi $

Where $ k $ is any integer.

For $ \cos(\theta) = -\sin(\theta) $:

$ \theta = \frac{3\pi}{4} + k \pi $

Where $ k $ is any integer.

Thus, the coordinates of the points on the unit circle are:

$ ( \cos(\frac{\pi}{4} + k \pi), \sin(\frac{\pi}{4} + k \pi) ) $

$ ( \cos(\frac{3\pi}{4} + k \pi), \sin(\frac{3\pi}{4} + k \pi) ) $

To determine the points on the unit circle that satisfy the equation $ cos^2( heta) – sin^2( heta) = 0 $:

We rewrite the equation: $ cos^2( heta) = sin^2( heta) $

Taking square roots: $ cos( heta) = pm sin( heta) $

For $ cos( heta) = sin( heta) $:

$ heta = frac{pi}{4} + k pi $

For $ cos( heta) = -sin( heta) $:

$ heta = frac{3pi}{4} + k pi $

Coordinates: $ ( cos(frac{pi}{4} + k pi), sin(frac{pi}{4} + k pi) ) $ and $ ( cos(frac{3pi}{4} + k pi), sin(frac{3pi}{4} + k pi) ) $

Given $ cos^2( heta) = sin^2( heta) $, solve $ cos( heta) = pm sin( heta) $:

$ heta = frac{pi}{4} + k pi $

and

$ heta = frac{3pi}{4} + k pi $

Coordinates: $ ( cos(frac{pi}{4} + k pi), sin(frac{pi}{4} + k pi) ) $ and $ ( cos(frac{3pi}{4} + k pi), sin(frac{3pi}{4} + k pi) ) $

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