For $a$ a real number, express $\cos (a), \sin (a)$ and $\tan (a)$ in terms of $t = \tan (a / 2)$. Deduce a rational parametrization of the circle, and provide a geometric interpretation; deduce a characterization of Pythagorean triplets.
Consider the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_10_8c44cd8b5c585c15e58bg-289.jpg?height=573&width=1065&top_left_y=336&top_left_x=336)

We have then \( O H = \cos(a), H M = \sin(a) \). We already know, thanks to the double-angle formulas, that

\[
\tan(a) = \frac{2t}{1 - t^2}
\]

Furthermore,

\[
\begin{aligned}
\frac{1}{t} & = \frac{1 + \cos(a)}{\sin(a)} \\
& = \frac{1}{\sin(a)} + \frac{1}{\tan(a)} \\
& = \frac{1}{\sin(a)} + \frac{1 - t^2}{2t}
\end{aligned}
\]

From this, we deduce the very important formulas:

\[
\begin{aligned}
\sin(a) & = \frac{2t}{1 + t^2} \\
\cos(a) & = \frac{1 - t^2}{1 + t^2} \\
\tan(a) & = \frac{2t}{1 - t^2}
\end{aligned}
\]

We observe that if \( t \) is rational, then \( \sin(a) \) and \( \cos(a) \) are also rational. This leads to the following rational parametrization [explain what this is]:

\[
\left\{\begin{array}{l}
x = \frac{1 - t^2}{1 + t^2} \\
y = \frac{2t}{1 + t^2}
\end{array}\right.
\]

This parametrization is called the stereographic projection. Indeed, in the figure, we have \( O C = t \), and the stereographic projection associates, to each point on the y-axis, a unique point on the circle. Thus, we have shown that this parametrization covers the entire circle except for the point \( S \).