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: $\frac{49}{28}$

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 : If $\theta$ is an acute angle and $\sin \theta+\operatorname{cosec} \theta=2$, then the value of $\sin ^5 \theta+\operatorname{cosec}^5 \theta$ is:

Option 1: 10

Option 2: 2

Option 3: 4

Option 4: 5

Question : If $\tan (11 \theta)=\cot (7 \theta)$, then what is the value of $\sin ^2(6 \theta)+\sec ^2(9 \theta)+\operatorname{cosec}^2(12 \theta) ?$

Option 1: $\frac{23}{6}$

Option 2: $\frac{35}{12}$

Option 3: $\frac{31}{12}$

Option 4: $\frac{43}{12}$

Question : If $\operatorname{cosec} 39°=x$, then the value of $\frac{1}{\operatorname{cosec}^{2}51°}+\sin^{2}39°+\tan^{2}51°-\frac{1}{\sin^{2}51°\sec^{2}39°}$ is:

Option 1: $\sqrt{x^{2}-1}$

Option 2: $\sqrt{1-x^{2}}$

Option 3: $x^{2}-1$

Option 4: $1-x^{2}$