Question : If $\tan3\theta \tan7\theta = 1$, where $7\theta$ is an acute angle, then find the value of $\cot15\theta$.

Option 1: $1$

Option 2: $-1$

Option 3: $-\sqrt{3}$

Option 4: $\sqrt{3}$

Question : $\theta$ is a positive acute angle and $\sin\theta-\cos\theta=0$, then the value of $\sec\theta+\operatorname{cosec}\theta$ is:

Option 1: $2$

Option 2: $\sqrt{3}$

Option 3: $2\sqrt{2}$

Option 4: $3\sqrt{2}$

Question : If $\cos^2 \theta=\frac{3}{4}$, where $\theta$ is an acute angle, then the value of $\sin \left(\theta+30^{\circ}\right)$ is:

Option 1: $1$

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

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

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

Question : If $\theta$ is a positive acute angle and $\tan2\theta \tan3\theta=1$, then the value of $2(\cos^2\frac{5\theta}{2}-1)$ is:

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

Option 2: -1

Option 3: 0

Option 4: $\frac{1}{2}$

Question : If $\tan\theta-\cot\theta=0$ and $\theta$ is positive acute angle, then the value of $\frac{\tan(\theta+15)}{\tan(\theta-15)}$ is:

Option 1: $3$

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

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

Option 4: $\sqrt{3}$

Question : If $\theta$ is a positive acute angle and $4\sin^{2}\theta =3$, then the value of $\left (\tan\theta-\cot\frac{\theta}{2}\right)$ is:

Option 1: $1$

Option 2: $0$

Option 3: $\sqrt{3}$

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