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 $\sqrt{\frac{1+\sin A}{1-\sin A}}$ 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 $\operatorname{cosec} A+\cot A=3$, $0 \leq A \leq 90^{\circ}$, then find the value of cos A.

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

Option 2: $\frac{2}{5}$

Option 3: $\frac{3}{5}$

Option 4: $\frac{4}{5}$

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 $\sec\theta+\tan\theta=2$, then the value of $\sec\theta$ is:

Option 1: $\frac{4}{5}$

Option 2: $5$

Option 3: $\frac{5}{4}$

Option 4: $\sqrt{2}$