Question : If $x\cos^{2}30^{\circ}\cdot \sin60^{\circ}=\frac{\tan^{2}45^{\circ}\cdot \sec60^{\circ}}{\operatorname{cosec}60^{\circ}}$, then the value of $x$ is:

Option 1: $\frac{1}{\sqrt{3}}$

Option 2: $\frac{1}{\sqrt{2}}$

Option 3: $2\frac{2}{3}$

Option 4: $\frac{1}{2}$

Question : $\left(\frac{\tan ^3 \theta}{\sec ^2 \theta}+\frac{\cot ^3 \theta}{\operatorname{cosec}^2 \theta}+2 \sin \theta \cos \theta\right) \div\left(1+\operatorname{cosec}^2 \theta+\tan ^2 \theta\right), 0^{\circ}<\theta<90^{\circ}$, is equal to:

Option 1: $\operatorname{cosec} \theta \sec \theta$

Option 2: $\operatorname{cosec} \theta$

Option 3: $\sin \theta \cos \theta$

Option 4: $\sec \theta$

Question : If $\operatorname{cosec}\theta+\sin\theta=\frac{5}{2}$, then the value of $(\operatorname{cosec}\theta-\sin\theta)$ is:

Option 1: $-\frac{3}{2}$

Option 2: $\frac{3}{2}$

Option 3: $-\frac{\sqrt{3}}{2}$

Option 4: $\frac{\sqrt{3}}{2}$

Question : Using $\operatorname{cosec}(\alpha+\beta)=\frac{\sec \alpha \times \sec \beta \times \operatorname{cosec} \alpha \times \operatorname{cosec} \beta}{\sec \alpha \times \operatorname{cosec} \beta+\operatorname{cosec} \alpha \times \sec \beta}$, find the value of $\operatorname{cosec} 75°$.

Option 1: $\frac{\sqrt{6}+\sqrt{2}}{4}$

Option 2: $\frac{\sqrt{6}-\sqrt{2}}{4}$

Option 3: $\sqrt{6}-\sqrt{2}$

Option 4: $\sqrt{6}+\sqrt{2}$

Question : If $3 \tan \theta=2 \sqrt{3} \sin \theta, 0^{\circ}<\theta<90^{\circ}$, then the value of $\frac{\operatorname{cosec}^2 2 \theta+\cot ^2 2 \theta}{\sin ^2 \theta+\tan ^2 2 \theta}$ is:

Option 1: $\frac{4}{13}$

Option 2: $\frac{20}{39}$

Option 3: $\frac{4}{3}$

Option 4: $\frac{20}{27}$