Question : In $\triangle{XYZ}$, right-angled at $Y$, if $\sin X = \frac{1}{2}$, find the value of $\cos X \cos Z + \sin X \sin Z$.

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

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

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

Option 4: $\sqrt{3}$

Question : If $\tan \frac{A}{2}=x$, then find $x$.

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

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

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

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

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 : If $x\sin^{3}\theta +y\cos^{3}\theta=\sin\theta\cos\theta$ and $x\sin\theta-y\cos\theta=0$, then the value of $\left ( x^{2}+y^{2} \right )$ equals:

Option 1: $1$

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

Option 3: $\frac{3}2$

Option 4: $2$

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