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 : Simplify the following:
$\frac{\cos x-\sqrt{3} \sin x}{2}$

Option 1: $\cos \left(\frac{\pi}{3}-x\right)$

Option 2: $\sin \left(\frac{\pi}{3}+x\right)$

Option 3: $\cos \left(\frac{\pi}{3}+x\right)$

Option 4: $\sin \left(\frac{\pi}{3}-x\right)$

Question : If $\frac{21\cos A+3\sin A}{3\cos A+4\sin A}=2$, then find the value of cot A.

Option 1: $\frac{9}{11}$

Option 2: $\frac{11}{9}$

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

Option 4: $\frac{11}{10}$

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 : What is the value of the given expression if $3\cot A=\frac{7}{3}$?
$\frac{3 \cos A+2 \sin A}{3 \cos A-2 \sin A}$

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

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

Option 3: $13$

Option 4: $1$