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Question
Prove the following trigonometric identities.
`1 + cot^2 theta/(1 + cosec theta) = cosec theta`
Solution
In the given question, we need to prove `1 + cot^2 theta/(1 + cosec theta) = cosec theta`
Using `cot theta = cos theta/sin theta` and `cosec theta = 1/sin theta` We get
`1 + cot^2 theta/(1 + cosec theta) = (1 = cosec theta + cot^2 theta)/(1 + cosec theta)`
`= ((1 + 1/sin theta + cos^2 theta/sin^2 theta))/((1 + 1/sin theta))`
` = (((sin^2 theta + sin theta + cos^2 theta)/sin^2 theta))/(((sin theta + 1)/sin theta))`
Further, using the property `sin^2 theta + cos^2 theta = 1`
We get
`((sin^2 theta + sin theta + cos^2 theta)/sin^2 theta)/((sin theta + 1)/sin theta) = ((1 + sin theta)/sin^2 theta)/((sin theta + 1)/sin theta)`
`= (1 + sin theta/sin^2 theta)((sin theta)/(1 + sin theta))`
`= 1/sin theta`
`= cosec theta`
Hence proved.
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