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

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embizze eNotes educator| Certified Educator

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.

embizze eNotes educator| Certified Educator

By squaring both sides you are assuming that the equality is true, but this is what was to be established.

Better is to multiply left side by sqrt(1-cos)/sqrt(1-cos) resulting in (1-cos)/sqrt(1-cos^2) or (1-cos)/sqrt(sin^2) which is (1-cos)/|sin| as required.