Question : If $\theta$ is a positive acute angles and $\operatorname{cosec}\theta =\sqrt{3}$, then the value of $\cot \theta -\operatorname{cosec}\theta$ is:

Option 1: $\sqrt2-\sqrt3$

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

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

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

Question : If $\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}=\frac{4}{5}$, then the value of $\frac{\operatorname{cosec}^2 \theta}{2-\operatorname{cosec}^2 \theta}$ is:

Option 1: $\frac{16}{25}$

Option 2: $\frac{40}{41}$

Option 3: $\frac{41}{40}$

Option 4: $\frac{31}{30}$

Question : If $\cos \theta+\sin \theta=\sqrt{2}$, then what is the value of $\sec \theta \operatorname{cosec} \theta$ ?

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

Option 2: $1$

Option 3: $2$

Option 4: $0$

Question : What is the value of $\sqrt{\frac{\operatorname{cosec} A+1}{\operatorname{cosec} A-1}}+\sqrt{\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}}$?

Option 1: $2 \cos A$

Option 2: $\sec A$

Option 3: $2\cos A$

Option 4: $2 \sec A$

Question : If $\sin \theta+\cos \theta=\frac{1}{29}$, then find the value of $\frac{\operatorname{sin} \theta+\operatorname{cos} \theta}{\operatorname{sin} \theta-\operatorname{cos} \theta}$.

Option 1: $\frac{1}{41}$

Option 2: $\frac{43}{29}$

Option 3: $\frac{41}{29}$

Option 4: $\frac{1}{43}$