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 : 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 $\alpha$ is an acute angle and $2\sin \alpha+15\cos^2\alpha=7$, then the value of $\cot \alpha$ is:

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

Option 2: $\frac{4}{5}$

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

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

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 $\sin A=\frac{\sqrt{3}}{2}, 0<A<90^{\circ}$, 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: $\frac{1}{\sqrt{3}}$