4.2. Determine, without using a calculator, the value of: \( \frac{\sin 45^{\circ} \cdot \operatorname{cosec} 90^{\circ}}{\sqrt{3} \cos 0^{\circ} \cdot \tan 60^{\circ}} \)

To determine the value of the expression \( \frac{\sin 45^{\circ} \cdot \operatorname{cosec} 90^{\circ}}{\sqrt{3} \cos 0^{\circ} \cdot \tan 60^{\circ}} \), we first substitute the known values of the trigonometric functions. 1. \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \) 2. \( \operatorname{cosec} 90^{\circ} = \frac{1}{\sin 90^{\circ}} = 1 \) 3. \( \cos 0^{\circ} = 1 \) 4. \( \tan 60^{\circ} = \sqrt{3} \) Now we can substitute these values into the expression: \[ \frac{\sin 45^{\circ} \cdot \operatorname{cosec} 90^{\circ}}{\sqrt{3} \cos 0^{\circ} \cdot \tan 60^{\circ}} = \frac{\left(\frac{\sqrt{2}}{2}\right) \cdot 1}{\sqrt{3} \cdot 1 \cdot \sqrt{3}} \] Simplifying the denominator: \[ \sqrt{3} \cdot \sqrt{3} = 3 \] So now, the expression becomes: \[ \frac{\frac{\sqrt{2}}{2}}{3} = \frac{\sqrt{2}}{2 \cdot 3} = \frac{\sqrt{2}}{6} \] Therefore, the value of the expression is \[ \frac{\sqrt{2}}{6}. \]