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 $\tan (11 \theta)=\cot (7 \theta)$, 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 the value of $\operatorname{cosec} A + \cot A = m$, then the value of $\operatorname{cosec} A - \cot A$ is:

Option 1: $\frac{1}m$

Option 2: $m$

Option 3: $\sqrt{m}$

Option 4: $m^2$

Question : If $7 \sin ^2 \theta+3 \cos ^2 \theta=4,0^{\circ}<\theta<90^{\circ}$, then the value of $(\tan ^2 2 \theta+\operatorname{cosec}^2 2 \theta)$ is:

Option 1: $7$

Option 2: $\frac{15}{4}$

Option 3: $\frac{13}{3}$

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

Question : If $\cot^2θ = 1 - e^2$, then the value of $\operatorname{cosec} θ + \cot^3θ \sec θ$ is:

Option 1: $\left(2-{e}^2\right)^ \frac{1}{2}$

Option 2: $\left(1-{e}^2\right)^ \frac{3}{2}$

Option 3: $\left(1-{e}^2\right)$

Option 4: $\left(2-{e}^2\right) ^\frac{3}{2}$