Question : If $a=45^{\circ}$ and $b=15^{\circ}$, what is the value of $\frac{\cos (a-b)-\cos (a+b)}{\cos (a-b)+\cos (a+b)} ?$

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

Option 2: $3-\sqrt{6}$

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

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

Question : Find the value of $\frac{\cos^2 15^{\circ}-\sin^2 15^{\circ}}{\cos^2 145^{\circ}+\sin^2 145^{\circ}}$.

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

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

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

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

Question : What is the value of $\frac{\tan 45^{\circ}-\tan 15^{\circ}}{1+\tan 45^{\circ} \tan 15^{\circ}}$?

Option 1: $\sqrt{3}$

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

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

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

Question : Using $\cos (A+B)=\cos A \cos B-\sin A \sin B$, find the value of $\cos 75^\circ$.

Option 1: $\frac{\sqrt{5}-1}{4}$

Option 2: $\frac{\sqrt{5}+1}{4}$

Option 3: $\frac{\sqrt{6}-\sqrt{2}}{4}$

Option 4: $\frac{\sqrt{6}+\sqrt{2}}{4}$

Question : Simplify $\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$

Option 1: $\frac{3 \sqrt{2}+\sqrt{6}}{8}$

Option 2: $\frac{\sqrt{3}}{2 \sqrt{2}-2 \sqrt{6}}$

Option 3: $\frac{3 \sqrt{2}-\sqrt{6}}{8}$

Option 4: $\frac{\sqrt{3}}{2 \sqrt{6}-2 \sqrt{2}}$

Question : What will be the value of $\frac{\sin 30^{\circ} \sin 40^{\circ} \sin 50^{\circ} \sin 60^{\circ}}{\cos 30^{\circ} \cos 40^{\circ} \cos 50^{\circ} \cos 60^{\circ}}$?

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

Option 2: $\sqrt{3}$

Option 3: $1$

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