Question : If $\frac{\tan\theta +\cot\theta }{\tan\theta -\cot\theta }=2, (0\leq \theta \leq 90^{0})$, then the value of $\sin\theta$ is:

Option 1: $\frac{2}{\sqrt3}$

Option 2: $\frac{\sqrt3}{2}$

Option 3: $\frac{1}{2}$

Option 4: $1$

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 $0 \leq \theta \leq 90^{\circ}$ and $\sec ^{107} \theta+\cos ^{107} \theta=2$, then, $(\sec \theta+\cos \theta)$ is equal to:

Option 1: $2$

Option 2: $1$

Option 3: $\frac{1}{2}$

Option 4: $2^{-107}$

Question : If $0°<\theta<90°$ and $2\sec\theta =3 \operatorname{cosec}^2 \theta$, then $\theta$ is:

Option 1: $\frac{\pi}{6}$

Option 2:

$\frac{\pi}{4}$

Option 3:

$\frac{\pi}{3}$

Option 4:

$\frac{\pi}{5}$

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: $\frac{2+\sqrt{2}}{2}$