Cosine of Angle plus Straight Angle/Proof 4

Proof
From the discussion in Sine and Cosine are Periodic on Reals:


 * $\map \sin {x + \eta} = \cos x$
 * $\map \cos {x + \eta} = -\sin x$

for $\eta \in \R_{>0}$, where $\pi$ was defined as $\pi := 2 \eta$.

It follows that $\eta = \dfrac \pi 2$, thus:


 * $\map \cos {x + \pi} = -\map \sin {x + \dfrac \pi 2} = -\cos x$