Answer
$-\sqrt2$
Work Step by Step
We know that $\sin$, $\csc$, $\tan$ are odd functions, which means $f(-\theta)=-f(\theta).$
We also know that $\cos$, $\sec$, are even functions, which means $f(-\theta)=f(\theta).$
Therefore,
$\csc{\left(-\frac{\pi}{4}\right)}\\
=-\csc{\left(\frac{\pi}{4}\right)}\\
=-\dfrac{1}{\sin(\frac{\pi}{4})}\\
=-\dfrac{1}{\frac{\sqrt2}{2}}\\
=-\dfrac{2}{\sqrt2}\\
=-\dfrac{2}{\sqrt2} \cdot \dfrac{\sqrt2}{\sqrt2}\\
=-\dfrac{2\sqrt2}{2}\\
=-\sqrt2$
