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 $\theta$ be an acute angle and $\tan \theta+\cot \theta=2$, then the value of $2 \tan ^2 \theta+\cot ^2 \theta+\tan ^4 \theta \cot ^4 \theta$ is:

Option 1: 4

Option 2: 2

Option 3: 3

Option 4: 6

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 $\tan (90-\theta)=\frac{2}{\sqrt{3}}$, then the value of $2 \sqrt{3} \tan \theta+1$ is:

Option 1: 4

Option 2: 5

Option 3: 3

Option 4: 6

Question : If $\sin\theta+\cos\theta=\sqrt{2}\cos\theta$, then the value of $\cot\theta$ is:

Option 1: $\sqrt{2}+1$

Option 2: $\sqrt{2}-1$

Option 3: $\sqrt{3}-1$

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