Cosine of Angle plus Straight Angle/Proof 4

Theorem

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

Proof

From the discussion in the proof of Real Cosine Function is Periodic:

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

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

for $\eta \in \R_{>0}$.

From Sine and Cosine are Periodic on Reals: Pi, we define $\pi \in \R$ as $\pi := 2 \eta$.

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

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

$\blacksquare$