Question

Prove that `1/(csctheta+1) + 1/(csctheta-1) -= 2sec theta \ tan theta`?

Answer 1

The identity as quoted is invalid, However:

`1/(csctheta+1) + 1/(csctheta-1) -= 2sec theta \ tan theta`

Explanation:

The identity as quoted is invalid, However:

We have:

`1/(csctheta+1) + 1/(csctheta-1) -= ((csctheta-1) + (csctheta+1))/((csctheta+1)(csctheta-1))`

`" " = (csctheta-1 + csctheta+1)/(csc^2theta+csctheta-csctheta-1)`

`" " = (2csctheta)/(csc^2theta-1)`

Using the trig identity `1+cot^2A -= csc^2A` we have:

`1/(csctheta+1) + 1/(csctheta-1) = (2csctheta)/(1+cot^2theta-1)`

`" " = (2csctheta)/(cot^2theta)`

`" " = (2csctheta)(tan^2theta)`

`" " = (2/sintheta)(sin theta)/(cos theta)tan theta`

`" " = 2/(cos theta)tan theta`

`" " = 2sec theta \ tan theta`