$\frac{{\sin \theta \cos \theta}}{1!}+\frac{{\sin 2\theta \cos ^2\theta}}{2!}+\frac{{\sin 3\theta \cos ^3\theta}}{3!}+...\infty$

Try $x\rightarrow \cos \theta \cdot e^{i\theta}$ in the same Maclaurin series. Then the imaginary part is what you are looking for.

$$e^{\cos\theta(\cos\theta + i\sin\theta)}=1+\frac{\cos\theta(\cos\theta + i\sin\theta)}{1!}+\frac{\cos^2\theta(\cos2\theta + i\sin2\theta)}{2!}+\frac{\cos^3\theta(\cos3\theta + i\sin3\theta)}{3!}+...$$