, 2022

Solution:
Note that if $\sin (\theta) = 0$, then the polynomial has 1 root. Now assume this is not the case then the polynomial is a quadratic in $x$.

Factor the polynomial as $(\tan (\theta) x + 1)(x + \sec (\theta))$. Then the condition is equivalent to $\sec (\theta) = \frac{1}{\tan (\theta)}$, which is equivalent to $\sin (\theta) = \cos^{2}(\theta) = 1 - \sin^{2}(\theta)$. Solving now gives $\sin (\theta) = \frac{\sqrt{5} - 1}{2}$ as the only solution.