Question : If $\sin A=\frac{\sqrt{3}}{2}, 0<A<90^{\circ}$, then find the value of $2(\operatorname{cosec} A + \cot A)$.

Option 1: $2 \sqrt{3}$

Option 2: $\sqrt{3}$

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

Option 4: $\frac{1}{\sqrt{3}}$

Question : If $\frac{1}{\operatorname{cosec} \theta+1}+\frac{1}{\operatorname{cosec} \theta-1}=2 \sec \theta, 0^{\circ}<\theta<90^{\circ}$, then the value of $\frac{\tan \theta+2 \sec \theta}{\operatorname{cosec} \theta}$ is:

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

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

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

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

Question : The value of $\frac{\sin A}{\cot A+\operatorname{cosec} A}-\frac{\sin A}{\cot A-\operatorname{cosec} A}+1$ is:

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

Option 2: $3$

Option 3: $0$

Option 4: $2$

Question : The value of $\sqrt{\frac{1+\sin A}{1-\sin A}}$ is:

Option 1: $\sec A-\tan A$

Option 2: $\operatorname{cosec} A+\cot A$

Option 3: $\sec A+\tan A$

Option 4: $\operatorname{cosec} A-\cot A$

Question : If $(\operatorname{cosec} \theta-\cot \theta) = \frac{7}{2}$, the value of $\operatorname{cosec} \theta$ is:

Option 1: $\frac{47}{28}$

Option 2: $\frac{51}{28}$

Option 3: $\frac{53}{28}$

Option 4: $\frac{49}{28}$