Equations that reduce to the standard forms

Solve $\sin\theta = \tan\theta$ by dividing both sides by $\sin\theta$: $1 = \frac{1}{\cos\theta}$, so $\cos\theta = 1$ and $\theta = 2n\pi$. But $\theta = \pi$ works too: $\sin\pi = 0 = \tan\pi$. Where did it go?