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Question
Prove the following trigonometric identities.
`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`
Solution
We know that, `sin^2 theta + cos^2 theta = 1`
Multiplying numerator and denominator under the square root by `1 - cos theta)` we have
`sqrt((1 - cos theta)/(1 + cos theta)) = sqrt(((1 - cos theta)(1 - cos theta))/((1 + cos theta)(1 - cos theta)))`
`= sqrt((1 - cos theta)^2/(1 - cos^2 theta))`
`= sqrt((1 - cos theta)^2/sin^2 theta`
`= (1 - cos theta)/sin theta`
`= 1/sin theta - cos theta/sin theta`
`= cosec theta - cot theta`
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