Question

Prove the following trigonometric identities.

`(tan^3 theta)/(1 + tan^2 theta) + (cot^3 theta)/(1 + cot^2 theta) = sec theta cosec theta - 2 sin theta cos theta`

Answer 1

`(tan^3 theta)/(1 + tan^2 theta) + (cot^3 theta)/(1 + cot^2 theta) `         [`∵ sec^2 theta - tan^2 theta = 1 - cosec^2 theta - cot^2 theta = 1`]

`= tan theta + cos^2 theta = cot^3 theta xx sin^3 theta`

`[∵ 1/sec^2 theta = cos^2 theta, 1/cosec^2 theta = 1 + cot^2 theta]`

`sin^3 theta/cos^3 theta xx cos^2 theta + cos^3 theta/sin^3 theta xx sin^2 theta`

`sin^3 theta/cos theta + cos^3 theta/sin theta`

`= (sin^4 theta + cos^4 theta)/(sin theta cos theta)`

` (1 - 2sin^2 theta cos^2 theta)/(sin theta cos theta)`

`1/(sin theta cos theta) - (2 sin^2 theta cos^2 theta)/(sin theta cos theta)`

`sec theta cosec theta - 2sin theta cos theta`.