Question : If $\frac{\cos ^2 \theta}{\cot ^2 \theta–\cos ^2 \theta}=3$, where $0^{\circ}<\theta<90^{\circ}$ then the value of $\theta$ is:

Option 1: $45^{\circ}$

Option 2: $50^{\circ}$

Option 3: $60^{\circ}$

Option 4: $30^{\circ}$

Question : If $\cot \theta=\frac{1}{\sqrt{3}}, 0^{\circ}<\theta<90^{\circ}$, then the value of $\frac{2-\sin ^2 \theta}{1-\cos ^2 \theta}+\left(\operatorname{cosec}^2 \theta-\sec \theta\right)$ is:

Option 1: 0

Option 2: 2

Option 3: 5

Option 4: 1

Question : If $3+\cos ^2 \theta=3\left(\cot ^2 \theta+\sin ^2 \theta\right), 0^{\circ}<\theta<90^{\circ}$, then what is the value of $(\cos \theta+2 \sin \theta)$ ?

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

Option 2: $3 \sqrt{2}$

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

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

Question : If $(\cos \theta+\sin \theta):(\cos \theta-\sin \theta)=(\sqrt{3}+1):(\sqrt{3}-1), 0^{\circ}<\theta<90^{\circ}$, then what is the value of $\sec \theta$?

Option 1: 2

Option 2: $\sqrt{2}$

Option 3: 1

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

Question : If $k\left(\tan 45^{\circ} \sin 60^{\circ}\right)=\cos 60^{\circ} \cot 30^{\circ}$, then the value of k is:

Option 1: $1$

Option 2: $2$

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

Option 4: $\sqrt{3}$