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 $\tan ^2 48^{\circ}-\operatorname{cosec}^2 42^{\circ}+\operatorname{cosec}\left(67^{\circ}+\theta\right)-\sec \left(23^{\circ}-\theta\right)$ is:

Option 1: $-1$

Option 2: $0$

Option 3: $1$

Option 4: $-2$

Question : If $\operatorname{cos} \theta+\operatorname{sin} \theta=\sqrt{2} \operatorname{cos} \theta$, find the value of $(\cos \theta-\operatorname{sin} \theta)$

Option 1: $\sqrt{2} \sin \theta$

Option 2: $\sqrt{2} \cos \theta$

Option 3: $\frac{1}{\sqrt{2}} \sin \theta$

Option 4: $\frac{1}{2}\cos \theta$

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 : 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$