Question : The value of $\frac{\sin ^2 30^{\circ}+\cos ^2 60^{\circ}+\sec 45^{\circ} × \sin 45^{\circ}}{\sec 60^{\circ}+{\text{cosec}} 30^{\circ}}$ is:

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

Option 2: $-\frac{3}{8}$

Option 3: $\frac{3}{8}$

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

Question : Find the value of $\sqrt{\frac{1-\tan A}{1+\tan A}}$.

Option 1: $\sqrt{\frac{1+\sin 2 A}{\cos 2 A}}$

Option 2: $\sqrt{\frac{1-\sin 2 A}{\cos 2 A}}$

Option 3: $\sqrt{\frac{1+\sin A}{\cos A}}$

Option 4: $\sqrt{\frac{1-\sin A}{\cos A}}$

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: $\sqrt{3}$

Question : If $4-2 \sin ^2 \theta-5 \cos \theta=0,0^{\circ}<\theta<90^{\circ}$, then the value of $\cos \theta-\tan \theta$ is:

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

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

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

Option 4: $\frac{1-2 \sqrt{3}}{2}$

Question : If $4-2 \sin ^2 \theta-5 \cos \theta=0,0^{\circ}<\theta<90^{\circ}$, then the value of $\cos \theta+\tan \theta$ is:

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

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

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

Option 4: $\frac{1-2 \sqrt{3}}{2}$