We have

${\text{L}}{\text{.H}}{\text{.S}}{\text{.}}$

$ = \frac{{1 + \cos 2x}}{2} + \frac{{1 + \cos \left( {2x + \frac{{2\pi }}{3}} \right)}}{2} + \frac{{1 + \cos \left( {2x – \frac{{2\pi }}{3}} \right)}}{2}$

$ = \frac{1}{2}\left[ {3 + \cos 2x + \cos \left( {2x + \frac{{2\pi }}{3}} \right) + \cos \left( {2x – \frac{{2\pi }}{3}} \right)} \right]$

$ = \frac{1}{2}\left[ {3 + \cos 2x + 2\cos 2x\cos \frac{{2\pi }}{3}} \right]$

$=\frac{1}{2}\left[3+\cos 2 x+2 \cos 2 x \cos \left(\pi-\frac{\pi}{3}\right)\right]$

$=\frac{1}{2}\left[3+\cos 2 x-2 \cos 2 x \cos \frac{\pi}{3}\right]$

$=\frac{1}{2}[3+\cos 2 x-\cos 2 x]=\frac{3}{2}= R.H.S.$