Question : If $\sec \theta+\tan \theta=\frac{1}{\sqrt{3}}$, then the positive value of $\cot \theta+\cos \theta$ is:

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

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

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

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

Question : If $\theta$ is an acute angle and $\sin \theta \cos \theta=2 \cos ^3 \theta-\frac{1}{4} \cos \theta$, then the value of $\sin \theta$ is:

Option 1: $\frac{\sqrt{15}-1}{8}$

Option 2: $\frac{\sqrt{15}-1}{4}$

Option 3: $\frac{\sqrt{15}+1}{4}$

Option 4: $\frac{\sqrt{15}-1}{2}$

Question : If $\sqrt{2} \sec ^2 \theta-4 \sec \theta+2 \sqrt{2}=0$, then what is the value $\sin ^2 \theta+\tan ^2 \theta$?

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

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

Option 3: $\frac{5}{2}$

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

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 : If $\theta$ is a positive acute angle and $4\cos ^{2}\theta -4\cos \theta +1=0$, then the value of $\tan (\theta -15^{\circ})$is equal to:

Option 1: $0$

Option 2: $1$

Option 3: $\sqrt{3}$

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