#### Answer

$$
\frac{d y}{d x}\frac{\cos t}{e^t}
$$
and at $t=20$, we have
$$
\frac{d y}{d x}= \frac{\cos 20}{e^{20}}.
$$

#### Work Step by Step

Since $x=e^t-1$ and $y=\sin t$, then we have
$$
\frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{y^{\prime}(t)}{x^{\prime}(t)}=\frac{\cos t}{e^t}
$$
and at $t=20$, we have
$$
\frac{d y}{d x}= \frac{\cos 20}{e^{20}}.
$$