Answer
$-cot(t) + cos(t) + C$
Work Step by Step
We can simplify the fraction in the integrand as $\int \frac{1}{sin^2(t)} - sin(t) dt = \int csc^2(t) - sin(t) dt$, by the trigonometric identity $ \frac{1}{sin(t)} = csc(t)$
Evaluating the integrand gives us $-cot(t) +cos(t) + C$
Note: Recall the trigonometric derivative identities $$ \frac{d}{dt} cot(t) = -csc^2(t) $$
and
$$ \frac{d}{dt} cos(t) = -sin(t) $$