Question : If $3\left(\cot ^2 \theta-\cos ^2 \theta\right)=1-\sin ^2 \theta, 0^{\circ}<\theta<90^{\circ}$, then $\theta$ is equal to:

Option 1: $30^{\circ}$

Option 2: $60^{\circ}$

Option 3: $45^{\circ}$

Option 4: $15^{\circ}$

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 : Find the value of $\frac{\sin ^2 39^{\circ}+\sin ^2\left(90^{\circ}–39^{\circ}\right)}{\cos ^2 35^{\circ}+\cos ^2\left(90^{\circ}–35^{\circ}\right)}+3 \tan 25^{\circ} \tan 75^{\circ}$:

Option 1: 2

Option 2: 4

Option 3: 3

Option 4: 1