ĐK: $\cos x \neq 0$ hay $x \neq \dfrac{\pi}{2} + k\pi$.

Ptrinh tương đương vs

$\tan^3x - 1 + \dfrac{1}{\cos^2x} + 2\dfrac{\cos(\dfrac{\pi}{3} - x)}{\sin(\dfrac{\pi}{3} - x)}=3$

Áp dụng tchat liên hệ giữa tan và cot và đẳng thức $\dfrac{1}{\cos^2x} = \tan^2x + 1$ ta có

$\tan^3x - 1 + \tan^2x + 1 + 2\dfrac{\dfrac{1}{2} \cos x + \dfrac{\sqrt{3}}{2} \sin x}{\dfrac{\sqrt{3}}{2} \cos x - \sin x \dfrac{1}{2}} = 3$

$<-> \tan^3x + \tan^2x - 3 + 2\dfrac{\cos x + \sqrt{3} \sin x}{\sqrt{3} \cos x - \sin x} = 0$

$<-> \tan^3x + \tan^2x - 3 - 2\dfrac{1 + \sqrt{3} \tan x}{\tan x- \sqrt{3}} = 0$

Nhân cả 2 vế vs $\tan x - \sqrt{3}$ ta có

$(\tan x - \sqrt{3})(\tan^3x + \tan^2x - 3) - 2(1 + \sqrt{3}\tan x) = 0$

$<-> \tan^4x + (1 - \sqrt{3}) \tan^3x -\tan^2x \sqrt{3} - (3 + 2\sqrt{3})\tan x + 3\sqrt{3} - 2 =0$

Nghiệm ở đây là

$\tan x = \dfrac{1}{4}(\sqrt{3} - 1)+ \dfrac{1}{2} \sqrt{1 - \dfrac{1}{\sqrt{3}} + \dfrac{\sqrt{3}}{2} - 10 \sqrt[3]{\dfrac{2}{3(180 + 211\sqrt{3} + \sqrt{308643-11556\sqrt{3}})}} + 11 \sqrt[6]{3}\sqrt[3]{\dfrac{2}{180 + 211\sqrt{3} + \sqrt{308643-11556\sqrt{3}}}} + \dfrac{\sqrt[3]{\dfrac{1}{2}(180 + 211\sqrt{3}+ \sqrt{308643-11556\sqrt{3}})}}{3^{\dfrac{2}{3}}}} -

\dfrac{1}{2} \sqrt{-1 + \dfrac{1}{\sqrt{3}} + \dfrac{3\sqrt{3}}{2} + \dfrac{3}{4}(1-\sqrt{3})^2+ 10 \sqrt[3]{\dfrac{2}{3(180 + 211\sqrt{3} + \sqrt{308643-11556\sqrt{3}})}} - 11 \sqrt[6]{3}\sqrt[3]{\dfrac{2}{180 + 211\sqrt{3} + \sqrt{308643-11556\sqrt{3}}}} - \dfrac{\sqrt[3]{\dfrac{1}{2}(180 + 211\sqrt{3}+ \sqrt{308643-11556\sqrt{3}})}}{3^{\dfrac{2}{3}}} + \dfrac{(1-\sqrt{3})(-4\sqrt{3}-(1-\sqrt{3})^2)-8(-3-2\sqrt{3})}{4 \sqrt{1 - \dfrac{1}{\sqrt{3}} + \dfrac{\sqrt{3}}{2} - 10 \sqrt[3]{\dfrac{2}{3(180 + 211\sqrt{3} + \sqrt{308643-11556\sqrt{3}})}} + 11 \sqrt[6]{3}\sqrt[3]{\dfrac{2}{180 + 211\sqrt{3} + \sqrt{308643-11556\sqrt{3}}}} + \dfrac{\sqrt[3]{\dfrac{1}{2}(180 + 211\sqrt{3}+ \sqrt{308643-11556\sqrt{3}})}}{3^{\dfrac{2}{3}}}}\sqrt{1 - \dfrac{1}{\sqrt{3}} + \dfrac{\sqrt{3}}{2} - 10 \sqrt[3]{\dfrac{2}{3(180 + 211\sqrt{3} + \sqrt{308643-11556\sqrt{3}})}} + 11 \sqrt[6]{3}\sqrt[3]{\dfrac{2}{180 + 211\sqrt{3} + \sqrt{308643-11556\sqrt{3}}}} + \dfrac{\sqrt[3]{\dfrac{1}{2}(180 + 211\sqrt{3}+ \sqrt{308643-11556\sqrt{3}})}}{3^{\dfrac{2}{3}}}}}}$