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 : $\cos \left(30^{\circ}+\theta\right)-\sin \left(60^{\circ}-\theta\right)=$ _____________.

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

Option 2: $0$

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

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

Question : If $\cos \left(2 \theta+54^{\circ}\right)=\sin \theta, 0^{\circ}<\left(2 \theta+54^{\circ}\right)<90^{\circ}$, then what is the value of $\frac{1}{\tan 5 \theta+\operatorname{cosec} \frac{5 \theta}{2}}$?

Option 1: $3\sqrt2$

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

Option 3: $2\sqrt3$

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

Question : If $\sin \theta-\cos \theta=0$, then find the value of $\left(\sin^3 \theta-\cos^3 \theta\right)$.

Option 1: $0$

Option 2: $2$

Option 3: $1$

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

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: $\sqrt{5}$