Question

What is `cot theta + tantheta*sectheta` in terms of `sintheta`?

Answer 1

`((1-sin^2theta)^(3/2) +sin^2theta)/(sintheta(1-sin^2theta)`

Explanation:

You need to use the facts that `cot theta = 1/tan theta` and that `tan theta = sin theta/cos theta`

Then
`cot theta +tan theta * sec theta = cos theta/sin theta + sin theta / cos theta * 1/cos theta`

We know that `sin^2theta + cos^2theta = 1` so

`cos^2theta = 1 - sin^2theta` and

`costheta = sqrt(1-sin^2theta)`

Substituting these into the expression gives

`sqrt(1-sin^2theta)/sintheta +sintheta/(1-sin^2theta)`

`=((1-sin^2theta)^(3/2) +sin^2theta)/(sintheta(1-sin^2theta)`