Biến đổi \[\sin x - \cos x\]

\[\begin{array}{l}
 \sin x - \cos x = \sqrt 2 .\frac{{\sqrt 2 }}{2}.\sin x - \sqrt 2 .\frac{{\sqrt 2 }}{2}.\cos x = \sqrt 2 .\left( {\frac{{\sqrt 2 }}{2}.\sin x - \frac{{\sqrt 2 }}{2}.\cos x} \right) \\
  = \sqrt 2 .\left( {\sin x.c{\rm{os}}\frac{\pi }{4} - \cos x.\sin \frac{\pi }{4}} \right) \\
  = \sqrt 2 .\sin \left( {x - \frac{\pi }{4}} \right) \\
 \end{array}\]
Miền giá trị của sinx - cosx:
Do:
\[ - 1 \le \sin \left( {x - \frac{\pi }{4}} \right) \le 1 \Rightarrow  - \sqrt 2  \le \sqrt 2 .\sin \left( {x - \frac{\pi }{4}} \right) \le \sqrt 2 \]

Tương tự, ta cũng có:
\[\begin{array}{l}
 \sin x - \cos x = \sqrt 2 .\frac{{\sqrt 2 }}{2}.\sin x - \sqrt 2 .\frac{{\sqrt 2 }}{2}.\cos x = \sqrt 2 .\left( {\frac{{\sqrt 2 }}{2}.\sin x - \frac{{\sqrt 2 }}{2}.\cos x} \right) \\
  =  - \sqrt 2 .\left( {\cos x.c{\rm{os}}\frac{\pi }{4} - \sin x.\sin \frac{\pi }{4}} \right) \\
  =  - \sqrt 2 .c{\rm{os}}\left( {x + \frac{\pi }{4}} \right) \\
 \end{array}\]

Mặt khác:

\[ - 1 \le c{\rm{os}}\left( {x + \frac{\pi }{4}} \right) \le 1 \Rightarrow  - \sqrt 2  \le  - \sqrt 2 .c{\rm{os}}\left( {x - \frac{\pi }{4}} \right) \le \sqrt 2 \]

Tóm lại:
\[\begin{array}{l}
 \sin x - \cos x = \left\{ \begin{array}{l}
 \sqrt 2 .\sin \left( {x - \frac{\pi }{4}} \right) \\
  - \sqrt 2 .c{\rm{os}}\left( {x + \frac{\pi }{4}} \right) \\
 \end{array} \right. \\
  - \sqrt 2  \le \sin x - \cos x \le \sqrt 2  \\
 \end{array}\]