`sqrt((1 - cos(theta))/(1 + cos(theta))) = (1 - cos(theta))/|sin(theta)|` Verfiy the identity.

Textbook Question

Chapter 5, 5.2 - Problem 42 - Precalculus (3rd Edition, Ron Larson).
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embizze | High School Teacher | (Level 2) Educator Emeritus

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Verify `sqrt((1-cos theta)/(1+cos theta))=(1-cos theta)/|sin theta|`

Start with the left side and show that it is equivalent to the right side:

`sqrt((1-cos theta)/(1+cos theta))`

`=sqrt(1-cos theta)/sqrt(1+cos theta)`

`=sqrt(1-cos theta)/sqrt(1+cos theta) * sqrt(1-cos theta)/sqrt(1-cos theta)`

`=(1-cos theta)/sqrt(1-cos^2 theta)`

`=(1-cos theta)/sqrt(sin^2 theta)`

`=(1-cos theta)/|sin theta|`

as required.

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