\(a)\,\,\tan \left( {2x + 1} \right)\tan \left( {3x - 1} \right) = 1\)
ĐK: \(\left\{ \matrix{ \cos \left( {2x + 1} \right) \ne 0 \hfill \cr \cos \left( {3x - 1} \right) \ne 0 \hfill \cr} \right.\)
\(\eqalign{ & pt \Leftrightarrow \tan \left( {2x + 1} \right) = {1 \over {\tan \left( {3x - 1} \right)}} = \cot \left( {3x - 1} \right) \cr & \Leftrightarrow \tan \left( {2x + 1} \right) = \tan \left( {{\pi \over 2} - 3x + 1} \right) \cr & \Leftrightarrow 2x + 1 = {\pi \over 2} - 3x + 1 + k\pi \cr & \Leftrightarrow 5x = {\pi \over 2} + k\pi \cr & \Leftrightarrow x = {\pi \over {10}} + {{k\pi } \over 5}\,\,\left( {k \in Z} \right)\,\,\left( {tm} \right) \cr} \)
Vậy nghiệm của phương trình là \(x = {\pi \over {10}} + {{k\pi } \over 5}\,\,\left( {k \in Z} \right)\).
\(b)\,\,\tan x + \tan \left( {x + {\pi \over 4}} \right) = 1\)
ĐK: \(\left\{ \matrix{ \cos x \ne 0 \hfill \cr \cos \left( {x + {\pi \over 4}} \right) \ne 0 \hfill \cr \tan x \ne 1 \hfill \cr} \right.\)
\(\eqalign{ & pt \Leftrightarrow \tan x + {{\tan x + 1} \over {1 - \tan x}} = 1 \cr & \Leftrightarrow \tan x - {\tan ^2}x + \tan x + 1 = 1 - \tan x \cr & \Leftrightarrow {\tan ^2}x - 3\tan x = 0 \cr & \Leftrightarrow \tan x\left( {\tan x - 3} \right) = 0 \cr & \Leftrightarrow \left[ \matrix{ \tan x = 0 \hfill \cr \tan x = 3 \hfill \cr} \right. \cr & \Leftrightarrow \left[ \matrix{ x = k\pi \hfill \cr x = \arctan 3 + k\pi \hfill \cr} \right.\,\,\,\left( {k \in Z} \right) (tm) \cr} \)
Vậy nghiệm của phương trình là \(x = k\pi \) hoặc \(x = \arctan 3 + k\pi \,\,\left( {k \in Z} \right)\).
