\( \frac{3 \tan 315^{\circ} \sin 225^{\circ} \sin 120^{\circ}}{\cos 315^{\circ} \cos 210^{\circ}} \)

1. Evaluate each trigonometric function: - \( \tan 315^\circ \): \(315^\circ = 360^\circ - 45^\circ\), so \(\tan 315^\circ = -\tan 45^\circ = -1\). - \( \sin 225^\circ \): \(225^\circ = 180^\circ + 45^\circ\), so \(\sin 225^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2}\). - \( \sin 120^\circ \): \(120^\circ = 180^\circ - 60^\circ\), so \(\sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2}\). - \( \cos 315^\circ \): \(315^\circ = 360^\circ - 45^\circ\), so \(\cos 315^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}\). - \( \cos 210^\circ \): \(210^\circ = 180^\circ + 30^\circ\), so \(\cos 210^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}\). 2. Substitute these values into the expression: \[ \frac{3 \tan 315^\circ \sin 225^\circ \sin 120^\circ}{\cos 315^\circ \cos 210^\circ} = \frac{3 \cdot (-1) \cdot \left(-\frac{\sqrt{2}}{2}\right) \cdot \frac{\sqrt{3}}{2}}{\frac{\sqrt{2}}{2} \cdot \left(-\frac{\sqrt{3}}{2}\right)} \] 3. Simplify the numerator: \[ 3 \cdot (-1) \cdot \left(-\frac{\sqrt{2}}{2}\right) \cdot \frac{\sqrt{3}}{2} = 3 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{6}}{4} \] 4. Simplify the denominator: \[ \frac{\sqrt{2}}{2} \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{\sqrt{6}}{4} \] 5. Form the complete fraction: \[ \frac{\frac{3\sqrt{6}}{4}}{-\frac{\sqrt{6}}{4}} = \frac{3\sqrt{6}}{4} \times \frac{-4}{\sqrt{6}} = -3 \] Thus, the value of the expression is \(-3\).