Sine of x minus Cosine of x/Sine Form

Theorem

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

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

Proof

\(\ds \sin x - \cos x\)

\(=\)

\(\ds \sin x - \map \sin {\frac \pi 2 - x}\)

Sine of Complement equals Cosine

\(\ds \)

\(=\)

\(\ds 2 \map \cos {\frac {x + \paren {\frac \pi 2 - x} } 2} \map \sin {\frac {x - \paren {\frac \pi 2 - x} } 2}\)

Sine minus Sine

\(\ds \)

\(=\)

\(\ds 2 \cos \frac \pi 4 \map \sin {x - \frac \pi 4}\)

simplifying

\(\ds \)

\(=\)

\(\ds \sqrt 2 \map \sin {x - \frac \pi 4}\)

Cosine of $\dfrac \pi 4$

$\blacksquare$