Evaluate the expressions without using a calculato \( 1 \quad 4 \tan 330^{\circ} \cos 135^{\circ} \sin 300^{\circ} \) \( 2 \quad \frac{3 \tan 315^{\circ} \sin 225^{\circ} \sin 120^{\circ}}{\cos 315^{\circ} \cos 210^{\circ}} \)

We start by evaluating each trigonometric function in the expressions. ### Expression 1 Evaluate \[ 4 \tan 330^{\circ} \cos 135^{\circ} \sin 300^{\circ}. \] 1. **Find \(\tan 330^{\circ}\):** Since \(330^{\circ} = 360^{\circ} - 30^{\circ}\), we have \[ \tan 330^{\circ} = -\tan 30^{\circ} = -\frac{1}{\sqrt{3}}. \] 2. **Find \(\cos 135^{\circ}\):** Note that \(135^{\circ} = 180^{\circ} - 45^{\circ}\), so \[ \cos 135^{\circ} = -\cos 45^{\circ} = -\frac{\sqrt{2}}{2}. \] 3. **Find \(\sin 300^{\circ}\):** Since \(300^{\circ} = 360^{\circ} - 60^{\circ}\), we get \[ \sin 300^{\circ} = -\sin 60^{\circ} = -\frac{\sqrt{3}}{2}. \] Substitute these into the expression: \[ 4 \cdot \left(-\frac{1}{\sqrt{3}}\right) \cdot \left(-\frac{\sqrt{2}}{2}\right) \cdot \left(-\frac{\sqrt{3}}{2}\right). \] Determine the sign: \(- \times - \times - = -\). Now compute the numerical part: \[ 4 \cdot \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}. \] Notice that \[ \frac{1}{\sqrt{3}} \cdot \sqrt{3} = 1. \] Thus, we have: \[ 4 \cdot \frac{\sqrt{2}}{2 \cdot 2} = 4 \cdot \frac{\sqrt{2}}{4} = \sqrt{2}. \] Incorporating the negative sign: \[ -\sqrt{2}. \] ### Expression 2 Evaluate \[ \frac{3 \tan 315^{\circ} \sin 225^{\circ} \sin 120^{\circ}}{\cos 315^{\circ} \cos 210^{\circ}}. \] 1. **Find \(\tan 315^{\circ}\):** Since \(315^{\circ} = 360^{\circ} - 45^{\circ}\), \[ \tan 315^{\circ} = -\tan 45^{\circ} = -1. \] 2. **Find \(\sin 225^{\circ}\):** With \(225^{\circ} = 180^{\circ} + 45^{\circ}\), \[ \sin 225^{\circ} = -\sin 45^{\circ} = -\frac{\sqrt{2}}{2}. \] 3. **Find \(\sin 120^{\circ}\):** Since \(120^{\circ} = 180^{\circ} - 60^{\circ}\), \[ \sin 120^{\circ} = \sin 60^{\circ} = \frac{\sqrt{3}}{2}. \] 4. **Find \(\cos 315^{\circ}\):** For \(315^{\circ} = 360^{\circ} - 45^{\circ}\), \[ \cos 315^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2}. \] 5. **Find \(\cos 210^{\circ}\):** Since \(210^{\circ} = 180^{\circ} + 30^{\circ}\), \[ \cos 210^{\circ} = -\cos 30^{\circ} = -\frac{\sqrt{3}}{2}. \] Substitute these into the expression: \[ \frac{3 \cdot (-1) \cdot \left(-\frac{\sqrt{2}}{2}\right) \cdot \left(\frac{\sqrt{3}}{2}\right)}{\left(\frac{\sqrt{2}}{2}\right) \cdot \left(-\frac{\sqrt{3}}{2}\right)}. \] **Simplify the numerator:** \[ 3 \cdot (-1) \cdot \left(-\frac{\sqrt{2}}{2}\right) = 3 \cdot \frac{\sqrt{2}}{2}, \] then multiplying by \(\frac{\sqrt{3}}{2}\) gives: \[ 3 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3 \sqrt{6}}{4}. \] **Simplify the denominator:** \[ \frac{\sqrt{2}}{2} \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{\sqrt{6}}{4}. \] Thus, the expression becomes: \[ \frac{\frac{3 \sqrt{6}}{4}}{-\frac{\sqrt{6}}{4}} = -3. \] ### Final Answers \[ \boxed{-\sqrt{2}} \quad \text{and} \quad \boxed{-3}. \]