(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 break down each part. For the first expression, we can use trigonometric identities. Notably, \(\sin(180^{\circ} - x) = \sin x\) and \(\sin(-x) = -\sin x\). Hence, the expression simplifies to \(\frac{\sin x + \sin x}{\sin(360^{\circ} + x)}\) which equals \(\frac{2\sin x}{\sin x} = 2\) (for \(x\) not equal to \(n\pi\), \(n \in \mathbb{Z}\)). Simplifying the denominator involves \(\sin(360^{\circ} + x) = \sin x\). For the second expression, we apply the cosine identity: \(\cos(720^{\circ} + \theta) = \cos \theta\) and \(\cos(-\theta) = \cos \theta\). Thus, the numerator simplifies to \(\cos \theta - \cos \theta = 0\), making the whole expression undefined as the denominator \(\tan(180^{\circ} - \theta) = -\tan \theta\) does not affect the outcome of the numerator being zero. So it results in the forms \(2\) and undefined respectively!