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 A is an acute angle, the simplified form of
$\frac{\cos (\pi-A) \cdot \cot \left(\frac{\pi}{2}+A\right) \cos (-A)}{\tan (\pi+A) \tan \left(\frac{3 \pi}{2}+A\right) \sin (2 \pi-A)}$ is:

Option 1: $ \cos^2 A$

Option 2: $\sin A$

Option 3: $\sin^2 A$

Option 4: $\cos A$

Question : If $\cos x+\sin x=\sqrt{2} \cos x$, what is the value of $(\cos x-\sin x)^2+(\cos x+\sin x)^2$?

Option 1: $2$

Option 2: $1$

Option 3: $0$

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

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 : If $\sec x- \cos x$ = 4, then what will be the value of $\frac{\left(1+\cos ^2x\right)}{\cos x}?$

Option 1: $\frac{9}{4}$

Option 2: $\frac{1}{4}$

Option 3: $2\sqrt{5}$

Option 4: $\sqrt{5}$