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

1. Start with the expression: \[ \frac{\cos (90^\circ + \theta)}{\sin (0^\circ - 180^\circ) + 3 \sin (-\theta)} \] 2. Simplify the numerator using the trigonometric identity: \[ \cos (90^\circ + \theta) = -\sin \theta \] Thus, the numerator becomes: \[ -\sin \theta \] 3. Simplify the first term of the denominator: \[ \sin (0^\circ - 180^\circ) = \sin (-180^\circ) \] Since \(\sin (-180^\circ) = -\sin 180^\circ\) and \(\sin 180^\circ = 0\), we have: \[ \sin (-180^\circ) = 0 \] So the denominator now is: \[ 0 + 3 \sin (-\theta) \] 4. Simplify the second term of the denominator using the identity: \[ \sin (-\theta) = -\sin \theta \] Therefore: \[ 3 \sin (-\theta) = 3 (-\sin \theta) = -3 \sin \theta \] 5. Now the expression becomes: \[ \frac{-\sin \theta}{-3 \sin \theta} \] 6. Cancel the common factor \(-\sin \theta\) (assuming \(\sin \theta \neq 0\)): \[ \frac{-\sin \theta}{-3 \sin \theta} = \frac{1}{3} \] 7. The simplified expression is: \[ \frac{1}{3} \]