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Question
Prove the following trigonometric identities.
`((1 + cot^2 theta) tan theta)/sec^2 theta = cot theta`
Solution
We have to prove `((1 + cot^2 theta) tan theta)/sec^2 theta = cot theta`
We know that, `sec^2 theta - tan^2 theta = 1`
So
`((1 + cot^2 theta)tan theta)/sec^2 theta = ((1 + cot^2 theta)tan theta)/(1 + tan^2 theta)`
` = ((1 + 1/tan^2 theta)tan theta)/(1 + tan^2 theta)`
`= (((tan^2 theta + 1)/(tan^2 theta)) tan theta)/(1 + tan^ 2 theta)`
`= ((1 + tan^2 theta)tan theta)/(tan^2 theta(1 + tan^2 theta))`
`= 1/tan theta`
`= cot theta`
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