Answer
$-1$
Work Step by Step
I know that $\csc$ has a period of $2\pi$, hence $\csc{\theta}=\tan{(\theta-2\pi)}.$ Thus $\csc{\frac{25\pi}{12}}=\csc{(\frac{25\pi}{12}-2\pi)}=\csc{\frac{\pi}{12}}$
I know that $\sin$ is an odd function, which means $f(-\theta)=-f(\theta).$ Therefore $\sin{-\frac{\pi}{12}}=-\sin{\frac{\pi}{12}}$.
Thus: $\csc{\frac{25\pi}{12}}\sin{-\frac{\pi}{12}}=\csc{\frac{\pi}{12}}-\sin{\frac{\pi}{12}}=-1$.
Because $\csc{\theta}\sin{\theta}=1$.
