Determine the value of \( \theta \) if \( \theta \in\left[0^{\circ}: 90^{\circ}\right] \), withou the use of a calculate \( \frac{\tan 30^{\circ} \cdot \operatorname{cosec} 60^{\circ}}{\cos 45^{\circ} \cdot \cot 45^{\circ}} \)

To determine the value of \( \theta \) in the expression \( \frac{\tan 30^{\circ} \cdot \operatorname{cosec} 60^{\circ}}{\cos 45^{\circ} \cdot \cot 45^{\circ}} \), we first evaluate each trigonometric function involved. 1. **Calculate \( \tan 30^{\circ} \)**: \[ \tan 30^{\circ} = \frac{1}{\sqrt{3}} \] 2. **Calculate \( \operatorname{cosec} 60^{\circ} \)**: \[ \operatorname{cosec} 60^{\circ} = \frac{1}{\sin 60^{\circ}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \] 3. **Calculate \( \cos 45^{\circ} \)**: \[ \cos 45^{\circ} = \frac{1}{\sqrt{2}} \] 4. **Calculate \( \cot 45^{\circ} \)**: \[ \cot 45^{\circ} = 1 \] Now substituting these values back into the expression: \[ \frac{\tan 30^{\circ} \cdot \operatorname{cosec} 60^{\circ}}{\cos 45^{\circ} \cdot \cot 45^{\circ}} = \frac{\left( \frac{1}{\sqrt{3}} \right) \cdot \left( \frac{2}{\sqrt{3}} \right)}{\left( \frac{1}{\sqrt{2}} \right) \cdot 1} \] Now let’s simplify the numerator and the denominator: 1. **Numerator**: \[ \frac{1}{\sqrt{3}} \cdot \frac{2}{\sqrt{3}} = \frac{2}{3} \] 2. **Denominator**: \[ \frac{1}{\sqrt{2}} \] Now we can express the entire fraction: \[ \frac{\frac{2}{3}}{\frac{1}{\sqrt{2}}} = \frac{2}{3} \cdot \sqrt{2} = \frac{2\sqrt{2}}{3} \] Now we have: \[ \frac{2\sqrt{2}}{3} \] Next, we need to determine \( \theta \) such that \( \tan \theta = \frac{2\sqrt{2}}{3} \). Since this fraction does not correspond to a standard angle, we will solve \( \theta = \tan^{-1}\left(\frac{2\sqrt{2}}{3}\right) \) giving us a specific angle \( \theta \) measured in degrees. So the final answer, given that \( \theta \) lies in the interval \( [0^\circ, 90^\circ] \), cannot be simplified further without a calculator, but it is clearly a positive angle within the given bounds. However, for a rough estimate, we might expect that \( \theta \) would be between \( 45^\circ \) and \( 60^\circ \), due to the values we calculated. Thus: \[ \theta \approx 45^\circ \text{ to } 60^\circ \] Exact calculation would necessitate further evaluation techniques or numerical methods.