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Question
Prove the following trigonometric identities.
`((1 + tan^2 theta)cot theta)/(cosec^2 theta) = tan theta`
Solution
We need to prove `((1 + tan^2 theta)cot theta)/(cosec^2 theta) = tan theta`
Solving the L.H.S, we get
`((1 + tan^2 theta)cot theta)/(cosec^2 theta) = (sec^2 theta (cot theta))/(cosec^2 theta)`
Using `sec theta = 1/cos theta, cot theta = cos theta/sin theta`. `cosec theta = 1/sin theta` we get
`(sec^2 theta(cot theta))/(cosec^2 theta) = (1/cos^2 theta (cos theta/sin theta))/(1/sin^2 theta)`
`= (1/(cos theta sin theta))/(1/sin^2 theta)`
`= sin^2 theta/(cos theta sin theta)`
`= sin theta/cos theta`
`= tan theta`
Hence proved.
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