Answer
$-\frac{4}{3}$
Work Step by Step
Step 1. Letting $sin^{-1}(-\frac{4}{5})=t$, we have $sin(t)=-\frac{4}{5}$ with $-\frac{\pi}{2}\lt t \lt 0$
Step 2. Form a right triangle of $y=4, r=5, x=3$ with angle $|t|$ facing $y$.
Step 3. We have $tan(sin^{-1}(-\frac{4}{5}))=tan(t)=-\frac{4}{3}$
