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

**(b) Simplify** 1. Rewrite the numerator using trigonometric identities: - \( \sin(180^\circ - x) = \sin x \) - \( \sin(-x) = -\sin x \) Thus, the numerator becomes: \[ \sin x - (-\sin x) = \sin x + \sin x = 2\sin x \] 2. Rewrite the denominator using the periodicity of sine: \[ \sin(360^\circ + x) = \sin x \] 3. The expression becomes: \[ \frac{2\sin x}{\sin x} \] 4. Cancel \(\sin x\) (assuming \(\sin x \neq 0\)): \[ 2 \] **(b) Answer:** \(2\) --- **(d) Simplify** 1. Rewrite the numerator using trigonometric identities: - Periodicity of cosine gives: \[ \cos(720^\circ + \theta) = \cos \theta \] - Cosine is an even function: \[ \cos(-\theta) = \cos \theta \] Thus, the numerator becomes: \[ \cos \theta - \cos \theta = 0 \] 2. Rewrite the denominator using the tangent identity: \[ \tan(180^\circ - \theta) = -\tan\theta \] 3. The expression becomes: \[ \frac{0}{-\tan \theta} \] 4. Since the numerator is \(0\) (assuming \(\tan\theta \neq 0\) so the denominator is defined), the entire expression is \(0\). **(d) Answer:** \(0\)