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

Question : For any acute angle $\theta, \sin \theta+\sin^2 \theta=1$, then the value of $\cos^2 \theta+\cos^4 \theta=$___________.

Option 1: 0

Option 2: 1

Option 3: 2

Option 4: –1

Question : If $\sin \theta=\frac{8}{17}$, where $\theta$ is an acute angle, then what is the value of $\tan \theta+\cot \theta ?$

Option 1: $\frac{217}{110}$

Option 2: $\frac{281}{190}$

Option 3: $\frac{289}{120}$

Option 4: $\frac{512}{321}$

Question : If $2 \cot \theta = 3$, find the value of $\frac{\sqrt{13} \sin \theta – 3 \tan \theta}{3 \tan \theta + \sqrt{13} \cos \theta}$

Option 1: $\frac{1}{\sqrt{13}}$

Option 2: $\frac{2}{\sqrt{13}}$

Option 3: 0

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