Question : If $\sqrt{3} \tan ^2 \theta-4 \tan \theta+\sqrt{3}=0$, then what is the value of $\tan ^2 \theta+\cot ^2 \theta$?

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

Option 2: $\frac{10}{3}$

Option 3: $3$

Option 4: $\frac{6}{5}$

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 $\tan\theta+\sec\theta=3$, $\theta$ being acute, the value of $5\sin\theta$ is:

Option 1: $\frac{5}{2}$

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

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

Option 4: $4$

Question : If $\tan \theta=\frac{8}{15}$, then the value of $\sqrt{\frac{1-\sin \theta}{1+\sin \theta}}$ is:

Option 1: $\frac{1}{5}$

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

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

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

Question :  If $\sec\theta +\tan\theta = 3$, then what is the value of $\sec \theta-\tan \theta$?

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

Option 2: $6$

Option 3: $\frac{1}{6}$

Option 4: $\frac{1}{5}$