The Pythagorean Identity for Sine and Cosine

In the setting of The Real Numbers: Standing Notation and Background, let $\cos$ and $\sin$ be the cosine and sine functions from $\mathbb{R}$ to $\mathbb{R}$, and let $t^{2}=tt$ as in The Real Numbers: Standing Notation and Background (i). Then the following hold for every $x\in\mathbb{R}$.

1. (The identity)(i)

$$(\cos x)^{2}+(\sin x)^{2}=1 .$$

2. (Bounds)(ii) $|\cos x|\le1$ and $|\sin x|\le1$.