\begin{aligned} \lim_{x \to \theta} \ \frac{x \sin \theta - \theta \sin x}{x-\theta} &= \lim_{x \to \theta} \ \frac{x \sin \theta - \theta \sin \theta + \theta \sin \theta - \theta \sin x}{x-\theta} \\[8pt] &= \lim_{x \to \theta} \ \frac{(x-\theta)\sin \theta + \theta (\sin \theta - \sin x) }{x-\theta} \\[8pt] &= \lim_{x \to \theta} \ \frac{(x-\theta)\sin \theta}{x-\theta} + \lim_{x \to \theta} \ \frac{\theta (\sin \theta - \sin x) }{x-\theta} \\[8pt] &= \sin \theta + \theta \cdot \lim_{x \to \theta} \ \frac{\sin \theta - \sin x}{x-\theta} \\[8pt] &= \sin \theta + \theta \cdot \lim_{x \to \theta} \ \frac{2 \cos \frac{1}{2}(\theta + x) \sin \frac{1}{2} (\theta-x)}{-2 \cdot \frac{1}{2}(\theta-x)} \\[8pt] &= \sin \theta - \theta \cdot \lim_{x \to \theta} \ \cos \frac{1}{2}(\theta + x) \cdot \lim_{x \to \theta} \ \frac{\sin \frac{1}{2} (\theta-x)}{\frac{1}{2}(\theta-x)} \\[8pt] &= \sin \theta - \theta \cdot \cos \theta \cdot 1 \\[8pt] &= \sin \theta - \theta \cos \theta \end{aligned}