Question

How do you simplify `(csc² t - cot² t) / sin² t`?

Answer 1

`sec^2 t`

Explanation:

I like just sines and cosines, so let's work with those:

`(csc^2 t - cot^2 t)/(sin^2(t))= (1/(sin^2t) - (cos^2t)/(sin^2t))/(sin^2t)`

Let's get rid of the fractions up top by multiplying top and bottom by `sin^2t`

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

Because `sin^2t + cos^2 t = 1`, `1- cos^2 t = sin^2 t`.
`= (sin^2t)/(sin^4t) = 1/(sin^2t) = sec^2 t`

In hindsight, we could have also realized that there is a trig identity that
`csc^2t-cot^2t = 1`
but that involves memorization, whereas the first thing doesn't.