simplify \( \frac{\sin 30^{\circ} \sin 60^{\circ}}{\cos 60^{\circ} \tan 60^{\circ}} \)

Let's break it down step by step! First, we need to recall the values of the trigonometric functions involved: - \( \sin 30^{\circ} = \frac{1}{2} \) - \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \) - \( \cos 60^{\circ} = \frac{1}{2} \) - \( \tan 60^{\circ} = \sqrt{3} \) Now we can substitute these values into the expression: \[ \frac{\sin 30^{\circ} \sin 60^{\circ}}{\cos 60^{\circ} \tan 60^{\circ}} = \frac{\left(\frac{1}{2}\right) \left(\frac{\sqrt{3}}{2}\right)}{\left(\frac{1}{2}\right) \left(\sqrt{3}\right)} \] Simplifying the numerator: \[ \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \] For the denominator: \[ \frac{1}{2} \cdot \sqrt{3} = \frac{\sqrt{3}}{2} \] Now putting the numerator and denominator together: \[ \frac{\frac{\sqrt{3}}{4}}{\frac{\sqrt{3}}{2}} = \frac{\sqrt{3}}{4} \cdot \frac{2}{\sqrt{3}} = \frac{2}{4} = \frac{1}{2} \] So, the simplified result is: \[ \frac{1}{2} \]