Sine of x plus Cosine of x/Sine Form

Theorem

$\sin x + \cos x = \sqrt 2 \sin \left({x + \dfrac \pi 4}\right)$

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

Proof

\(\ds \sin x + \cos x\)

\(=\)

\(\ds \sqrt 2 \cos \left({x - \dfrac \pi 4}\right)\)

Sine of x plus Cosine of x: Cosine Form

\(\ds \)

\(=\)

\(\ds \sqrt 2 \sin \left({x - \frac \pi 4 + \frac \pi 4}\right)\)

Sine of Angle plus Right Angle

\(\ds \)

\(=\)

\(\ds \sqrt 2 \sin \left({x + \frac \pi 4}\right)\)

simplifying

$\blacksquare$