(b) \( \frac{\sin \left(180^{\circ}-x\right)-\sin (-x)}{\sin \left(360^{\circ}+x\right)} \) (d) \( \frac{\cos \left(720^{\circ}+\theta\right)-\cos (-\theta)}{\tan \left(180^{\circ}-\theta\right)} \)

Let's dive into some fun trigonometric properties! The expression \( \sin(180^\circ - x) = \sin(x) \) simplifies the numerator significantly, while \( \sin(-x) = -\sin(x) \) adds a twist that changes the dynamics. Combining these, the numerator becomes \( \sin(x) + \sin(x) = 2\sin(x) \). On the denominator, \( \sin(360^\circ + x) = \sin(x) \), giving us a neat result of \( \frac{2\sin(x)}{\sin(x)} = 2 \), provided \( x \) is not a multiple of \( 180^\circ \). In the second expression, similar reductions occur. The cosine function is periodic; hence \( \cos(720^\circ + \theta) = \cos(\theta) \) and \( \cos(-\theta) = \cos(\theta) \). This means the numerator becomes \( \cos(\theta) - \cos(\theta) = 0 \). The denominator \( \tan(180^\circ - \theta) = -\tan(\theta) \). When everything is plugged in, we end up with \( \frac{0}{-\tan(\theta)} = 0 \), since the numerator is zero—trigonometric magic at its finest!