Question

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

Answer 1

`1 - cos^2 theta - 1 / cos theta + 1 / (1 - cos^2 theta)`

Explanation:

You should use the following identities:

[1] `" "sin^2 theta + cos^2 theta = 1 " "<=>" " sin^2 theta = 1 - cos^2 theta`

[2] `" "sec theta = 1 / cos theta`

[3] `" "csc theta = 1 / sin theta`

Thus, your expression can be transformed as follows:

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

`= 1 - cos^2 theta - 1 / cos theta + 1 / sin^2 theta`

.... use `sin^2 theta = 1 - cos^2 theta` once again...

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