Question : Which of the following is the value of $\sqrt{\frac{1-\sin 45^{\circ}}{1+\sin 45^{\circ}}}$?

Option 1: $\cos 45^{\circ} - \tan 45^{\circ}$

Option 2: $\tan 45^{\circ} - \sec 45^{\circ}$

Option 3: $\tan 45^{\circ}$

Option 4: $\sec 45^{\circ} - \tan 45^{\circ}$

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 : What is the value of $\frac{\sin (A+B)}{\sin A \cos B}$?

Option 1: $1 + \cot A \tan B$

Option 2: $1 + \tan A \cot B$

Option 3: $1 – \sin A \cos B$

Option 4: $1 − \cot A \tan B$

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

Option 1: 2

Option 2: 4

Option 3: 3

Option 4: 1