Question

How do you prove `(csc(-t)-sin(-t))/sin(t) = -cot^2(t)`?

Answer 1

Remember that `sin` is an odd function so `sin(-t)=-sin(t)` and that `cot=cos/sin` and `csc=1/sin` so you get:

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

Where I used the fact that: `sin^2(t)+cos^2(t)=1`
and: `cos^2(t)=1-sin^2(t)`
so `-cos^2(t)=-1+sin^2(t)`