Simplify the following: (a) \( \frac{\cos \left(180^{\circ}-\theta\right) \cdot \cos \left(90^{\circ}-\theta\right)}{\sin \left(90^{\circ}+\theta\right) \cdot \sin \left(-\theta-180^{\circ}\right)} \)

To simplify the expression: \[ \frac{\cos \left(180^{\circ}-\theta\right) \cdot \cos \left(90^{\circ}-\theta\right)}{\sin \left(90^{\circ}+\theta\right) \cdot \sin \left(-\theta-180^{\circ}\right)} \] Let's break it down step by step using trigonometric identities. 1. **Simplify \(\cos(180^\circ - \theta)\):** \[ \cos(180^\circ - \theta) = -\cos \theta \] 2. **Simplify \(\cos(90^\circ - \theta)\):** \[ \cos(90^\circ - \theta) = \sin \theta \] 3. **Simplify \(\sin(90^\circ + \theta)\):** \[ \sin(90^\circ + \theta) = \cos \theta \] 4. **Simplify \(\sin(-\theta - 180^\circ)\):** \[ \sin(-\theta - 180^\circ) = \sin(-(\theta + 180^\circ)) = -\sin(\theta + 180^\circ) \] Using the identity \(\sin(\theta + 180^\circ) = -\sin \theta\): \[ \sin(-\theta - 180^\circ) = -(-\sin \theta) = \sin \theta \] 5. **Substitute the simplified terms back into the original expression:** \[ \frac{(-\cos \theta) \cdot \sin \theta}{\cos \theta \cdot \sin \theta} \] 6. **Cancel out the common terms \(\cos \theta\) and \(\sin \theta\) in the numerator and denominator:** \[ \frac{-\cos \theta \cdot \sin \theta}{\cos \theta \cdot \sin \theta} = -1 \] **Final Answer:** \(-1\)