Sine of x minus Cosine of x/Cosine Form

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Theorem

$\sin x - \cos x = \sqrt 2 \, \map \cos {x - \dfrac {3 \pi} 4}$

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


Proof

\(\ds \sin x - \cos x\) \(=\) \(\ds \sqrt 2 \, \map \sin {x - \dfrac \pi 4}\) Sine of x minus Cosine of x: Sine Form
\(\ds \) \(=\) \(\ds \sqrt 2 \, \map \cos {\frac \pi 2 - \paren {x - \dfrac \pi 4} }\) Cosine of Complement equals Sine
\(\ds \) \(=\) \(\ds \sqrt 2 \, \map \cos {\frac \pi 2 - x + \dfrac \pi 4}\) simplifying
\(\ds \) \(=\) \(\ds \sqrt 2 \, \map \cos {\frac {3 \pi} 4 - x}\) simplifying
\(\ds \) \(=\) \(\ds \sqrt 2 \, \map \cos {x - \frac {3 \pi} 4}\) Cosine Function is Even

$\blacksquare$