Solve the equation giving the exact solutions which lie in $\left[0, 2\pi \right)$: $\;$ $\;$ $\cos \left(5x\right) \; \cos \left(3x\right) - \sin \left(5x\right) \; \sin \left(3x\right) = \dfrac{\sqrt{3}}{2}$
$\cos \left(5x\right) \; \cos \left(3x\right) - \sin \left(5x\right) \; \sin \left(3x\right) = \dfrac{\sqrt{3}}{2}$
i.e. $\;$ $\cos \left(5 x + 3 x\right) = \dfrac{\sqrt{3}}{2}$
i.e. $\;$ $\cos \left(8 x\right) = \dfrac{\sqrt{3}}{2} = \cos \left(\dfrac{\pi}{6}\right)$
i.e. $\;$ $8 x = 2 n \pi \pm \dfrac{\pi}{6}, \;\;\; n \in Z$
i.e. $\;$ $x = \dfrac{n \pi}{4} \pm \dfrac{\pi}{48}, \;\;\; n \in Z$
In the range $\left[0, 2 \pi \right)$,
when $\;$ $n = 0$, $\;$ $x = \dfrac{\pi}{48}$
when $\;$ $n = 1$, $\;$ $x = \dfrac{\pi}{4} - \dfrac{\pi}{48} = \dfrac{11 \pi}{48}$ $\;$ OR $\;$ $x = \dfrac{\pi}{4} + \dfrac{\pi}{48} = \dfrac{13 \pi}{48}$
when $\;$ $n = 2$, $\;$ $x = \dfrac{\pi}{2} - \dfrac{\pi}{48} = \dfrac{23 \pi}{48}$ $\;$ OR $\;$ $x = \dfrac{\pi}{2} + \dfrac{\pi}{48} = \dfrac{25 \pi}{48}$
when $\;$ $n = 3$, $\;$ $x = \dfrac{3 \pi}{4} - \dfrac{\pi}{48} = \dfrac{35 \pi}{48}$ $\;$ OR $\;$ $x = \dfrac{3 \pi}{4} + \dfrac{\pi}{48} = \dfrac{37 \pi}{48}$
when $\;$ $n = 4$, $\;$ $x = \pi - \dfrac{\pi}{48} = \dfrac{47 \pi}{48}$ $\;$ OR $\;$ $x = \pi + \dfrac{\pi}{48} = \dfrac{49 \pi}{48}$
when $\;$ $n = 5$, $\;$ $x = \dfrac{5 \pi}{4} - \dfrac{\pi}{48} = \dfrac{59 \pi}{48}$ $\;$ OR $\;$ $x = \dfrac{5 \pi}{4} + \dfrac{\pi}{48} = \dfrac{61 \pi}{48}$
when $\;$ $n = 6$, $\;$ $x = \dfrac{3\pi}{2} - \dfrac{\pi}{48} = \dfrac{71 \pi}{48}$ $\;$ OR $\;$ $x = \dfrac{3\pi}{2} + \dfrac{\pi}{48} = \dfrac{73 \pi}{48}$
when $\;$ $n = 7$, $\;$ $x = \dfrac{7 \pi}{4} - \dfrac{\pi}{48} = \dfrac{83 \pi}{48}$ $\;$ OR $\;$ $x = \dfrac{7 \pi}{4} + \dfrac{\pi}{48} = \dfrac{85 \pi}{48}$
when $\;$ $n = 8$, $\;$ $x = 2 \pi - \dfrac{\pi}{48} = \dfrac{95 \pi}{48}$
$\therefore \;$ For the given equation, the exact solutions which lie in $\left[0, 2\pi \right)$ are:
$x = \dfrac{\pi}{48}$, $\;$ $\dfrac{11\pi}{48}$, $\;$ $\dfrac{13\pi}{48}$, $\;$ $\dfrac{23\pi}{48}$, $\;$ $\dfrac{25\pi}{48}$, $\;$ $\dfrac{35\pi}{48}$, $\;$ $\dfrac{37\pi}{48}$, $\;$ $\dfrac{47\pi}{48}$, $\;$ $\dfrac{49\pi}{48}$, $\;$ $\dfrac{59\pi}{48}$, $\;$ $\dfrac{61\pi}{48}$, $\;$ $\dfrac{71\pi}{48}$, $\;$ $\dfrac{73\pi}{48}$, $\;$ $\dfrac{83\pi}{48}$, $\;$ $\dfrac{85\pi}{48}$, $\;$ $\dfrac{95\pi}{48}$