TITLE: trigonometry with alternative parametrizations of the circle
QUESTION [19 upvotes]: In trigonometry it is conventional when defining $\cos\theta$, $\sin\theta$, etc., to parametrize the circle by arc length $\theta$.  Some trigonometric identities don't depend at all on which parametrization of the circle is used, e.g. $\cos^2\theta+\sin^2\theta=1$.  For others that choice of parametrization is crucial, e.g. $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$.  To me it's amazing how MUCH follows from the simple identities taught to 15-year-olds, things like this and this and this and this, and if you want something yet more sophisticated, this item bearing a famous name.  By current stereotype the term "trigonometric identity" connotes something rather humble, but look at that last item.
In a previous question I asked about the role of addition and subtraction.  But now I will pose a different question:
Does anything similarly elaborate (the variety of different sorts of identities, some simple; others more involved; some that could bear famous names) happen with any other parametrizations of the circle?
Perhaps the most notable among other parametrizations --- almost surely one of the most well known --- is this:
$$
\begin{align}
x & = \frac{1-t^2}{1+t^2} \\[8pt]
y & = \frac{2t}{1+t^2}
\end{align}
$$
(The parameter space should be the one-point compactification $\mathbb{R}\cup\{\infty\}$.)

REPLY [12 votes]: If we let $\mathop{\mathrm{cosr}}(t) = \frac{1-t^2}{1+t^2}$ and $\mathop{\mathrm{sinr}}(t) = \frac{2t}{1+t^2}$ and of course $\mathop{\mathrm{tanr}}(t) = \frac{2t}{1-t^2}$ (short for "rational cosine" and "rational sine"), then there is some kind of trigonometric identity between $\mathop{\mathrm{cosr}}$ and $\mathop{\mathrm{sinr}}$ that resembles $\cos(\alpha+\beta) = \cos\alpha\,\cos\beta - \sin\alpha\,\sin\beta$ and $\sin(\alpha+\beta) = \cos\alpha\,\sin\beta + \sin\alpha\,\cos\beta$, namely:
$$\mathop{\mathrm{cosr}}(u\star v) = \mathop{\mathrm{cosr}}u\,\mathop{\mathrm{cosr}}v - \mathop{\mathrm{sinr}}u\,\mathop{\mathrm{sinr}}v$$
$$\mathop{\mathrm{sinr}}(u\star v) = \mathop{\mathrm{cosr}}u\,\mathop{\mathrm{sinr}}v + \mathop{\mathrm{sinr}}u\,\mathop{\mathrm{cosr}}v$$
and also
$$\mathop{\mathrm{tanr}}(u\star v) = \mathop{\mathrm{tanr}}u \star \mathop{\mathrm{tanr}}v$$
where
$$u \star v = \frac{u+v}{1-uv}$$
is the rational function giving composition of tangents ($\tan(\alpha+\beta) = \tan\alpha \star \tan\beta$).  This is simply because $\mathop{\mathrm{cosr}}(\tan\frac{\theta}{2}) = \cos\theta$ and similarly $\mathop{\mathrm{sinr}}(\tan\frac{\theta}{2}) = \sin\theta$.  Also note that $\mathop{\mathrm{tanr}}t = t\star t$.
The moral of the story is that $\star$ defines the group law on the circle when parametrized using $t$ (precisely, $\star$ is an algebraic group law on $\mathbb{P}^1_{\mathbb{R}} \setminus \{\pm i\}$, the latter being isomorphic to the circle using $t \mapsto (\mathop{\mathrm{cosr}}t,\mathop{\mathrm{sinr}}t)$).
[Edit: Here is a slightly different way of putting it.  We should see $\cos$ as the composite of two functions: first, the transcendental parametrization $\theta \mapsto (\cos\theta,\sin\theta)$ of the circle (or $\theta \mapsto e^{i\theta}$ if you want), and second the projection of the circle to the $x$ coordinate; similarly, $\sin$ is the same with the projection to the $y$ coordinate, while the above defined $\mathop{\mathrm{cosr}}$ and $\mathop{\mathrm{sinr}}$ replace the transcendental parametrization by the rational parametrization which is an isomorphism between $\mathbb{P}^1_{\mathbb{R}} \setminus \{\pm i\}$ and the circle.  Now an identity such as $\cos(\alpha+\beta) = \cos\alpha\,\cos\beta - \sin\alpha\,\sin\beta$ consists of two parts: (A) the transcendental parametrization $\theta \mapsto (\cos\theta,\sin\theta)$ is an homomorphism of (complex analytic) groups from $\mathbb{R}$ (with its usual addition) to the circle (with its usual composition, e.g., multiplication of complex numbers of modulus $1$), and (B) if $x,y$ are the (projections to the) first and second coordinates of the circle, then composition on the circle is given by $x = x'x'' - y'y''$ and $y = x'y'' + y'x''$ if $(x,y)$ is the composite of $(x',y')$ and $(x'',y'')$.  For the rational parametrization, we need to change (A) to say that the rational parametrization is an isomorphism groups between $\mathbb{P}^1_{\mathbb{R}} \setminus \{\pm i\}$ and the circle, where the latter has its usual composition and the former has… whatever composition law makes the rational parametrization an isomorphism(!), and which when computed gives the group law $\star$ written above.]