Cosine of Angle plus Straight Angle/Proof 1

Theorem

$\map \cos {x + \pi} = -\cos x$

Proof

\(\ds \map \cos {x + \pi}\)

\(=\)

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

Cosine of Sum

\(\ds \)

\(=\)

\(\ds \cos x \cdot \paren {-1} - \sin x \cdot 0\)

Cosine of Straight Angle and Sine of Straight Angle

\(\ds \)

\(=\)

\(\ds -\cos x\)

$\blacksquare$