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 $\tan(5\theta-10^\circ)=\cot(5\varphi+20^\circ)$, then the value of $(\theta+\varphi)$ is:

Option 1: $16^\circ$

Option 2: $18^\circ$

Option 3: $15^\circ$

Option 4: $20^\circ$

Question : If $\tan \theta+\sin \theta=A$ and $\tan \theta-\sin \theta=B$, then what is the value of $A^2-B^2$?

Option 1: $\tan \theta \sin \theta$

Option 2: $4 \cot \theta$

Option 3: $4 \tan \theta \sin \theta$

Option 4: $2 \tan \theta \sin \theta$

Question : If $\tan \theta \cdot \tan 2 \theta=1$, then the value of $\cot 5 \theta$ is:

Option 1: $-1$

Option 2: $1$

Option 3: $-\sqrt{3}$

Option 4: $\sqrt{3}$

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: $\frac{6}{5}$