Problem (Algebra):
If $\cos \theta=\tan \theta$, determine all possible values of $\sin \theta$, giving your answer(s) as simplified exact numbers.

Solution:
Since $\tan \theta=\frac{\sin \theta}{\cos \theta}$, then we assume that $\cos \theta \neq 0$.

Therefore, we obtain the following equivalent equations:

$$
\begin{aligned}
\cos \theta & =\tan \theta \\
\cos \theta & =\frac{\sin \theta}{\cos \theta} \\
\cos ^{2} \theta & =\sin \theta \\
1-\sin ^{2} \theta & =\sin \theta \\
0 & =\sin ^{2} \theta+\sin \theta-1
\end{aligned}
$$

Let $u=\sin \theta$. This quadratic equation becomes $u^{2}+u-1=0$

By the quadratic formula, $u=\frac{-1 \pm \sqrt{1^{2}-4(1)(-1)}}{2(1)}=\frac{-1 \pm \sqrt{5}}{2}$.

Therefore, $\sin \theta=\frac{-1+\sqrt{5}}{2} \approx 0.62$ or $\sin \theta=\frac{-1-\sqrt{5}}{2} \approx-1.62$.

Since $-1 \leq \sin \theta \leq 1$, then the second solution is inadmissible. Thus, $\sin \theta=\frac{-1+\sqrt{5}}{2}$.

Final answer: $\frac{-1+\sqrt{5}}{2}$