Simplizy \( \frac{\cos (90+\theta)}{\sin \left(0-180^{\circ}\right)+3 \sin (-\theta)} \)

Let’s start by simplifying the expression step by step. First, we know: \[ \cos(90^\circ + \theta) = -\sin(\theta) \] Now, let's look at the denominator: \[ \sin(0 - 180^\circ) = \sin(-180^\circ) = 0 \] Thus, the denominator simplifies to: \[ 0 + 3 \sin(-\theta) = 3 \sin(-\theta) = -3 \sin(\theta) \quad \text{(using } \sin(-x) = -\sin(x)\text{)} \] Now, we can substitute these values into the original expression: \[ \frac{\cos(90+\theta)}{\sin(0-180^{\circ}) + 3 \sin(-\theta)} = \frac{-\sin(\theta)}{-3 \sin(\theta)} \] Now simplifying this gives: \[ \frac{-\sin(\theta)}{-3 \sin(\theta)} = \frac{1}{3} \] as long as \(\sin(\theta) \neq 0\). So, the final simplified result is: \[ \frac{1}{3} \]