Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\tan x.tan\left( {45^\circ + x} \right) - \tan \left( {45^\circ + x} \right) + \tan x\\
= \tan \left( {45^\circ + x} \right)\left( {\tan x - 1} \right) + \tan x\\
= \frac{{\sin \left( {45^\circ + x} \right)}}{{\cos \left( {45^\circ + x} \right)}}\left( {\tan x - 1} \right) + \tan x\\
= \frac{{\sin 45^\circ .\cos x + \cos 45^\circ .\sin x}}{{\cos 45^\circ .\cos x - \sin 45^\circ .\sin x}}\left( {\tan x - 1} \right) + \tan x\\
= \frac{{\frac{{\sqrt 2 }}{2}.\cos x + \frac{{\sqrt 2 }}{2}\sin x}}{{\frac{{\sqrt 2 }}{2}\cos x - \frac{{\sqrt 2 }}{2}\sin x}}\left( {\frac{{\sin x}}{{\cos x}} - 1} \right) + \tan x\\
= \frac{{\cos x + \sin x}}{{\cos x - \sin x}}.\frac{{\sin x - \cos x}}{{\cos x}} + \frac{{\sin x}}{{\cos x}}\\
= \frac{{\cos x + \sin x}}{{ - \cos x}} + \frac{{\sin x}}{{\cos x}}\\
= \frac{{\cos x + \sin x - \sin x}}{{ - \cos x}}\\
= \frac{{\cos x}}{{ - \cos x}} = - 1
\end{array}\)
