Question

How do you express `4 cos^2 theta - sec^2 theta + 2 cot theta` in terms of `sin theta`?

Answer 1

`4 - 4 sin^2 theta - 1 / (1 - sin^2 theta) + (2 sqrt(1 - sin^2 theta)) / sin theta`

Explanation:

We will use the following identities:

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

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

[3] `" " cot theta = cos theta / sin theta`

Thus, we can express the term as follows:

`4 cos^2 theta - sec^2 theta + 2 cot theta`

`= 4(1 - sin^2 theta) - 1 / cos^2 theta + (2 cos theta) / sin theta`

... apply [1] once again...

`= 4 - 4 sin^2 theta - 1 / (1 - sin^2 theta) + (2 cos theta) / sin theta`

... now, there is only one `cos theta` expression left which we can express as `sqrt(1 - sin^2 theta)`:

`= 4 - 4 sin^2 theta - 1 / (1 - sin^2 theta) + (2 sqrt(1 - sin^2 theta)) / sin theta`

Hope that this helped!