Sine of x plus Cosine of x/Cosine Form

Theorem

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

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

\(\ds \)

\(=\)

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

Sine plus Sine

\(\ds \)

\(=\)

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

simplifying

\(\ds \)

\(=\)

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

Sine of $\dfrac \pi 4$

$\blacksquare$