Question

How do you express `cos theta - cos^2 theta + sec^2 theta` in terms of `sin theta`?

Answer 1

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

Explanation:

We use the Pythagorean identity `sin^2 theta + cos^2 theta = 1`

`=> cos^2 theta = 1 - sin^2 theta`
`=> cos theta = sqrt(1 - sin^2 theta)`
`=> sec^2 theta = 1/(cos^2 theta) = 1/(1 - sin^2 theta)`

`therefore cos theta - cos^2 theta + sec^2 theta = sqrt(1 - sin^2 theta) + 1 - sin^2 theta + 1/(1 - sin^2 theta)`