Cosine of x minus Sine of x/Cosine Form

Theorem

$\cos x - \sin x = \sqrt 2 \, \map \cos {x + \dfrac \pi 4}$

where $\sin$ denotes sine and $\cos$ denotes cosine.

$\ds \cos x - \sin x$

$=$

$\ds \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}$

$\ds $

$=$

$\ds \sqrt 2 \, \map \cos {\frac \pi 2 - \paren {x + \dfrac {3 \pi} 4} }$

$\ds $

$=$

$\ds \sqrt 2 \, \map \cos {\frac \pi 2 - x - \dfrac {3 \pi} 4}$

simplifying

$\ds $

$=$

$\ds \sqrt 2 \, \map \cos {-\frac \pi 4 - x}$

simplifying

$\ds $

$=$

$\ds \sqrt 2 \, \map \cos {x + \frac \pi 4}$

$\blacksquare$