\(\left( {2\sin x - 1} \right)\left( {\sqrt 3 \tan x + 2\sin x} \right) = 3 - 4{\cos ^2}x\,\,\left( * \right)\)

Điều kiện: \(\cos x \ne 0 \Leftrightarrow x \ne \dfrac{\pi }{2} + k\pi \).

\(\begin{array}{l}\left( * \right) \Leftrightarrow \left( {2\sin x - 1} \right).\dfrac{{\sqrt 3 \sin x + 2\sin x\cos x}}{{\cos x}} = 3 - 4{\cos ^2}x\\ \Leftrightarrow \left( {2\sin x - 1} \right)\left( {\sqrt 3 \sin x + \sin 2x} \right) + \left( {4{{\cos }^3}x - 3\cos x} \right) = 0\\ \Leftrightarrow 2\sqrt 3 {\sin ^2}x - \sqrt 3 \sin x + 2\sin x\sin 2x - \sin 2x + \cos 3x = 0\\ \Leftrightarrow 2\sqrt 3 {\sin ^2}x - \sqrt 3 \sin x + \cos x - \cos 3x - \sin 2x + \cos 3x = 0\\ \Leftrightarrow \sqrt 3 \sin x\left( {2\sin x - 1} \right) - \sin 2x + \cos x = 0\\ \Leftrightarrow \sqrt 3 \sin x\left( {2\sin x - 1} \right) - \cos x\left( {2\sin x - 1} \right) = 0\\ \Leftrightarrow \left( {2\sin x - 1} \right)\left( {\sqrt 3 \sin x - \cos x} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l}2\sin x - 1 = 0\,\,\left( 1 \right)\\\sqrt 3 \sin x - \cos x = 0\,\,\left( 2 \right)\end{array} \right.\end{array}\)

Giải \(\left( 1 \right) \Leftrightarrow \sin x = \dfrac{1}{2} \Leftrightarrow \left[ \begin{array}{l}x = \dfrac{\pi }{6} + k2\pi \\x = \dfrac{{5\pi }}{6} + k2\pi \end{array} \right.\).

Giải \(\left( 2 \right) \Leftrightarrow \sqrt 3 \sin x = \cos x \Leftrightarrow \sqrt 3 \tan x = 1 \Leftrightarrow \tan x = \dfrac{1}{{\sqrt 3 }} \Leftrightarrow x = \dfrac{\pi }{6} + k\pi \left( {TM} \right)\).

Hợp nghiệm của \(\left( 1 \right)\) và \(\left( 2 \right)\) ta được \(\left[ \begin{array}{l}x = \dfrac{\pi }{6} + k\pi \\x = \dfrac{{5\pi }}{6} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)\).

Mà \(x \in \left[ {0;20\pi } \right] \Rightarrow x \in \left\{ {\dfrac{\pi }{6};\dfrac{\pi }{6} + \pi ;...;\dfrac{\pi }{6} + 19\pi ;\dfrac{{5\pi }}{6};\dfrac{{5\pi }}{6} + 2\pi ;...\dfrac{{5\pi }}{6} + 18\pi } \right\}\)

Vậy tổng các nghiệm là:

\(\begin{array}{l}\,\,\,\,\,\dfrac{\pi }{6} + \dfrac{\pi }{6} + \pi + \dfrac{\pi }{6} + 2\pi + ... + \dfrac{\pi }{6} + 19\pi + \dfrac{{5\pi }}{6} + \dfrac{{5\pi }}{6} + 2\pi + ... + \dfrac{{5\pi }}{6} + 18\pi \\ = 20.\dfrac{\pi }{6} + \left( {1 + 2 + 3 + ... + 19} \right)\pi + \dfrac{{5\pi }}{6}.10 + 2\pi \left( {1 + 2 + ... + 9} \right) = \dfrac{{875\pi }}{3}\end{array}\).