Question : If $r\sin\theta=1$, $r\cos\theta=\sqrt{3}$, then the value of $(\sqrt{3}\tan\theta+1)$ is:

Option 1: $\sqrt{3}$

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

Option 3: $1$

Option 4: $2$

Question : If $\cos \theta+\cos ^2 \theta=1$, find the value of $\sqrt{\sin ^4 \theta+\cos ^2 \theta}$.

Option 1: $\sqrt{2} \cos \theta$

Option 2: $2 \operatorname{cos} \theta$

Option 3: $\sqrt{2} \operatorname{sin} \theta$

Option 4: $2 \operatorname{sin} \theta$

Question : If $\tan ^2 \theta+\tan ^4 \theta=1$, then:

Option 1: $\cot ^2 \theta+\cot ^4 \theta=1$

Option 2: $\cos ^2 \theta+\cos ^4 \theta=1$

Option 3: $\sin ^2 \theta+\sin ^4 \theta=1$

Option 4: $\operatorname{cosec}^2 \theta+\sec ^4 \theta=1$

Question : If $\frac{\sin ^2 \theta}{\cos ^2 \theta-3 \cos \theta+2}=1, \theta$ lies in the first quadrant, then the value of $\frac{\tan ^2 \frac{\theta}{2}+\sin ^2 \frac{\theta}{2}}{\tan \theta+\sin \theta}$ is:

Option 1: $\frac{2 \sqrt{3}}{27}$

Option 2: $\frac{5 \sqrt{3}}{27}$

Option 3: $\frac{2 \sqrt{3}}{9}$

Option 4: $\frac{7 \sqrt{3}}{54}$

Question : If $7 \sin ^2 \theta+4 \cos ^2 \theta=5$ and $\theta$ lies in the first quadrant, then what is the value of $\frac{\sqrt{3} \sec \theta+\tan \theta}{\sqrt{2} \cot \theta-\sqrt{3} \cos \theta}$?

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

Option 2: $3 \sqrt{2}$

Option 3: $2(\sqrt{2}-1)$

Option 4: $4 \sqrt{2}$