Sine of x minus Cosine of x/Cosine Form
Jump to navigation
Jump to search
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 |
$\blacksquare$