Question

Prove that:

`tan theta/(1 - cot theta) + cot theta/(1 - tan theta)` = 1 + sec θ cosec θ

Answer 1

LHS = `tan theta/(1 - cot theta) + cot theta/(1 - tan theta)`

= `tan theta/(1 - 1/tan theta) + (1/tan theta)/(1 - tan theta)`

= `tan^2theta/(tan theta - 1) + 1/(tan theta(1 - tan theta)`

= `tan^2theta/(tan theta - 1) - 1/(tan theta(tan theta - 1)`

= `(tan^3theta - 1)/(tan theta(tan theta - 1))`

= `((tan theta - 1)(tan^2theta + tan theta + 1))/(tan theta(tan theta - 1))`      ....[Using a³ − b³ = (a − b) (a² + ab + b²]

= `(tan^2theta + tan theta + 1)/(tan theta)`

= `(sec^2theta + tan theta)/(tan theta) = sec^2/tan theta + 1`

= `cos theta/(cos^2theta sin theta) + 1`

= sec θ . cosec θ + 1

= 1 + sec θ . cosec θ

= RHS

Hence Proved.