Answer
$-1$
Work Step by Step
Recall:
Sine is an odd function therefore $\sin{(-\theta)} = - \sin{\theta}$.
This means that:
$\sin{\left(-\dfrac{\pi}{12} \right)} = - \sin{\left(\dfrac{\pi}{12} \right)} $
Also,
$\sin{\left(-\dfrac{\pi}{12} \right)} \cdot \csc{\left(\dfrac{25 \pi }{12} \right)} = - \sin{\left(\dfrac{\pi}{12} \right)} \cdot \csc{\left(\dfrac{25 \pi }{12} \right)}$
Recal that:
$\sin{(\theta+2 \pi)} = \sin{\theta}$
Thus,
$\sin{\left(\dfrac{\pi}{12} \right)} = \sin{\left(\dfrac{\pi}{12}+2\pi \right)} = \sin{\left(\dfrac{25 \pi}{12} \right)}$
Therefore,
$-\sin{\left(\dfrac{\pi}{12} \right)} \cdot \csc{\left(\dfrac{25 \pi }{12} \right)} =- \sin{\left(\dfrac{25 \pi}{12} \right)}\cdot \csc{\left(\dfrac{25 \pi }{12} \right)}$
Recall:
$\csc{\theta} =\dfrac{1}{\sin{\theta}}$
Hence,
$-\sin{\left(\dfrac{25 \pi}{12} \right)}\cdot \csc{\left(\dfrac{25 \pi }{12} \right)} =- \sin{\left(\dfrac{25 \pi}{12} \right)}\cdot \dfrac{1}{ \sin{\left(\dfrac{25 \pi}{12} \right)}} = \boxed{-1}$
