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 $\cot \theta=\frac{1}{\sqrt{3}}, 0^{\circ}<\theta<90^{\circ}$, then the value of $\frac{2-\sin ^2 \theta}{1-\cos ^2 \theta}+\left(\operatorname{cosec}^2 \theta-\sec \theta\right)$ is:

Option 1: 0

Option 2: 2

Option 3: 5

Option 4: 1

Question : Find the value of $\left(\tan ^2 \theta+\tan ^4 \theta\right)$.

Option 1: $\cot ^2 \theta-\tan ^2 \theta$

Option 2: $\ {\sec}^4 \theta-\ {\sec}^2 \theta$

Option 3: $\ {\sec}^4 \theta-\ {\sec}^4 \theta$

Option 4: $ \ {\sec}^4 \theta+\ {\sec}^2 \theta$

Question : Solve the following equation.
$\sec ^2 \theta\left(\sqrt{1-\sin ^2 \theta}\right)= $ __________.

Option 1: $\tan \theta$

Option 2: $\operatorname{cosec \theta}$

Option 3: $\sec \theta$

Option 4: $1$

Question : The value of $\left(2 \cos ^2 \theta-1\right)\left[\frac{1+\tan \theta}{1-\tan \theta}+\frac{1-\tan \theta}{1+\tan \theta}\right]$ is:

Option 1: $2$

Option 2: $0$

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

Option 4: $1$