Putnam 1996 B4

For any square matrix $A$, we can define $\sin A$ by the usual power series: $$\sin A = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} A^{2n+1}.$$ Prove or disprove: there exists a $2 \times 2$ matrix $A$ with real entries such that $$\sin A = \left( \begin{array}{cc} 1 & 1996 \\ 0 & 1 \end{array} \right).$$