Question

How do we prove that `(1 - tan theta)/(1 + tan theta) = (cot theta - 1)/(cot theta + 1)`?

Answer 1

Use `tantheta = sintheta/costheta` and `cottheta = costheta/sintheta`.

`(1 - sintheta/costheta)/(1 +sintheta/costheta) = (costheta/sintheta - 1)/(costheta/sintheta + 1)`

`((costheta - sin theta)/costheta)/((costheta+ sin theta)/costheta) = ((costheta - sin theta)/sintheta)/((costheta + sin theta)/sintheta)`

`(costheta - sin theta)/(costheta + sin theta) = (costheta - sin theta)/(costheta + sin theta)`

This is true for all values of `theta`, so we have proved this identity.

Hopefully this helps!