Answer
$-e^{\csc(\pi+t)} +c$
Work Step by Step
Solve $\int e^{\csc(\pi+t)} \csc(\pi+t) \cot(\pi+t)dt$
Let us $p=\csc(\pi+t)$ and $dp=-\csc(\pi+t) \cot(\pi+t)dt$
This implies $\int e^{\csc(\pi+t)} \csc(\pi+t) \cot(\pi+t)dt=-\int e^p dp$
$=- e^p +c$
$=-e^{\csc(\pi+t)} +c$
Hence,$\int e^{\csc(\pi+t)} \csc(\pi+t) \cot(\pi+t)dt=-e^{\csc(\pi+t)} +c$