$\begin{array}{l}
\sin x + \cos x\\
= \sqrt 2 \left( {\dfrac{{\sqrt 2 }}{2}\sin x + \dfrac{{\sqrt 2 }}{2}\cos x} \right)\\
= \sqrt 2 \left( {\cos \dfrac{\pi }{4}\sin x + \sin \dfrac{\pi }{4}\cos x} \right) = \sqrt 2 \sin \left( {x + \dfrac{\pi }{4}} \right)\\
= \sqrt 2 \left( {\cos x\cos \dfrac{\pi }{4} + \sin x\sin \dfrac{\pi }{4}} \right) = \sqrt 2 \cos \left( {x - \dfrac{\pi }{4}} \right)\\
\sin x - \cos x\\
= \sqrt 2 \left( {\dfrac{{\sqrt 2 }}{2}\sin x - \dfrac{{\sqrt 2 }}{2}\cos x} \right) = \sqrt 2 \left( {\sin x\cos \dfrac{\pi }{4} - \cos x\sin \dfrac{\pi }{4}} \right)\\
= \sqrt 2 \sin \left( {x - \dfrac{\pi }{4}} \right)\\
= - \sqrt 2 \left( {\cos x\cos \dfrac{\pi }{4} - \sin x.\sin \dfrac{\pi }{4}} \right) = - \sqrt 2 \cos \left( {x + \dfrac{\pi }{4}} \right)
\end{array}$
