Phương trình $\tan x + \tan \left( {x + \frac{\pi }{3}} \right) + \tan \left( {x + \frac{{2\pi }}{3}} \right) = 3\sqrt 3 $ tương đương với phương trình:
A. $\cot x = \sqrt 3 .$
B. $\cot 3x = \sqrt 3 .$
C. $\tan x = \sqrt 3 .$
D. $\tan 3x = \sqrt 3 .$
A. $\cot x = \sqrt 3 .$
B. $\cot 3x = \sqrt 3 .$
C. $\tan x = \sqrt 3 .$
D. $\tan 3x = \sqrt 3 .$
Chọn C.
Trước hết, ta lưu ý công thức nhân ba: $\sin 3a = 3\sin a - 4{\sin ^3}a$; $\cos 3a = 4{\cos ^3}a - 3\cos a$; $\tan 3a = \frac{{3\tan a - {{\tan }^3}a}}{{1 - 3{{\tan }^2}a}}$.
$PT \Leftrightarrow \tan x + \frac{{\tan x + \tan \frac{\pi }{3}}}{{1 - \tan x\tan \frac{\pi }{3}}} + \frac{{\tan x + \tan \frac{{2\pi }}{3}}}{{1 - \tan x\tan \frac{{2\pi }}{3}}} = 3\sqrt 3 \Leftrightarrow \tan x + \frac{{\tan x + \sqrt 3 }}{{1 - \sqrt 3 \tan x}} + \frac{{\tan x - \sqrt 3 }}{{1 + \sqrt 3 \tan x}} = 3\sqrt 3 $
$ \Leftrightarrow \frac{{\tan x\left( {1 - 3{{\tan }^2}x} \right) + \left( {\tan x + \sqrt 3 } \right)\left( {1 + \sqrt 3 \tan x} \right) + \left( {\tan x - \sqrt 3 } \right)\left( {1 - \sqrt 3 \tan x} \right)}}{{1 - 3{{\tan }^2}x}} = 3\sqrt 3 $
$ \Leftrightarrow \frac{{\tan x - 3{{\tan }^3}x + \tan x + \sqrt 3 {{\tan }^2}x + \sqrt 3 + 3\tan x + \tan x - \sqrt 3 {{\tan }^2}x - \sqrt 3 + 3\tan x}}{{1 - 3{{\tan }^2}x}} = 3\sqrt 3 $
$ \Leftrightarrow \frac{{9\tan x - 3{{\tan }^3}x}}{{1 - 3{{\tan }^2}x}} = 3\sqrt 3 \Leftrightarrow \frac{{3\tan x - {{\tan }^3}x}}{{1 - 3{{\tan }^2}x}} = \sqrt 3 \Leftrightarrow \tan 3x = \sqrt 3 .$
Trước hết, ta lưu ý công thức nhân ba: $\sin 3a = 3\sin a - 4{\sin ^3}a$; $\cos 3a = 4{\cos ^3}a - 3\cos a$; $\tan 3a = \frac{{3\tan a - {{\tan }^3}a}}{{1 - 3{{\tan }^2}a}}$.
$PT \Leftrightarrow \tan x + \frac{{\tan x + \tan \frac{\pi }{3}}}{{1 - \tan x\tan \frac{\pi }{3}}} + \frac{{\tan x + \tan \frac{{2\pi }}{3}}}{{1 - \tan x\tan \frac{{2\pi }}{3}}} = 3\sqrt 3 \Leftrightarrow \tan x + \frac{{\tan x + \sqrt 3 }}{{1 - \sqrt 3 \tan x}} + \frac{{\tan x - \sqrt 3 }}{{1 + \sqrt 3 \tan x}} = 3\sqrt 3 $
$ \Leftrightarrow \frac{{\tan x\left( {1 - 3{{\tan }^2}x} \right) + \left( {\tan x + \sqrt 3 } \right)\left( {1 + \sqrt 3 \tan x} \right) + \left( {\tan x - \sqrt 3 } \right)\left( {1 - \sqrt 3 \tan x} \right)}}{{1 - 3{{\tan }^2}x}} = 3\sqrt 3 $
$ \Leftrightarrow \frac{{\tan x - 3{{\tan }^3}x + \tan x + \sqrt 3 {{\tan }^2}x + \sqrt 3 + 3\tan x + \tan x - \sqrt 3 {{\tan }^2}x - \sqrt 3 + 3\tan x}}{{1 - 3{{\tan }^2}x}} = 3\sqrt 3 $
$ \Leftrightarrow \frac{{9\tan x - 3{{\tan }^3}x}}{{1 - 3{{\tan }^2}x}} = 3\sqrt 3 \Leftrightarrow \frac{{3\tan x - {{\tan }^3}x}}{{1 - 3{{\tan }^2}x}} = \sqrt 3 \Leftrightarrow \tan 3x = \sqrt 3 .$