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Question
Prove the following trigonometric identities.
`(cos theta)/(cosec theta + 1) + (cos theta)/(cosec theta - 1) = 2 tan theta`
Solution
In the given question, we need to prove `(cos theta)/(cosec theta + 1) + (cos theta)/(cosec theta - 1) = 2 tan theta`
Using the identity `a^2 - b^2 = (a + b)(a - b)`
`cos theta/((cosec theta + 1)) + cos theta/(cosec theta - 1) = (cos theta(cosec theta - 1)+ cos theta(cosec theta + 1))/(cosec^2 theta - 1)`
`= (cos theta (cosec theta - 1 + cosec theta + 1))/(cosec^2 theta -1) = (cos theta(2 cosec theta))/cot^2 theta`
`= ((2 cos theta)(1/sin theta))/((cos^2 theta/sin^2 theta))`
`= 2 ((cos theta)/(sin theta))(sin^2 theta/cos^2 theta)`
`= 2 sin theta/cos theta`
`= 2 tan theta`
Hence proved.
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