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 §numbers. Then the following hold for every $x\in\mathbb{R}$.

1. (The identity)¶

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