This was a very surprising discovery for me that identities like this exist:
$$\tan \frac{c}{2}=\tan \frac{a}{2}\tan \frac{b}{2} \qquad \rightarrow$$
$$\tanh^{-1} (\cos c)=\tanh^{-1} (\cos a)+\tanh^{-1} (\cos b)$$
This is a fairly well known one, and can be proven by making substitutions:
$$u=\tan \frac{a}{2}, \qquad v=\tan \frac{b}{2}$$
Another, interesting one exists (proven in the same way):
$$\tan \frac{c}{2}=\frac{\tan \frac{a}{2}-\tan \frac{b}{2}}{\tan \frac{a}{2}+\tan \frac{b}{2}} \qquad \rightarrow$$
$$\tanh^{-1} (\sin c)=\tanh^{-1} (\cos b)-\tanh^{-1} (\cos a)$$
What other identities like this exist?
What is the interpretation of such identities in terms of:
- Complex numbers
- Geometry
Or is it just a coincidense with no particular significance?
Not sure if this is the sort of thing you're after.
$\displaystyle \tanh^{-1}(y) = \frac1{2} \ln \left( \frac{1+y}{1-y} \right)$
Using $\displaystyle \cos(x) = 2\cos^2 \left( \frac{x}{2} \right) - 1$ you get:
$\displaystyle \tanh^{-1}(\cos(x)) = -\ln \left( \tan\left( \frac{x}{2} \right) \right)$
So the $\displaystyle \tanh^{-1}(\cos(x))$ terms have logarithmic properties wrt $\displaystyle \tan \left( \frac{x}{2} \right)$.
This explains the sum of $\displaystyle \tanh^{-1}$ terms producing a product in $\displaystyle \tan$ terms.
It also implies that more terms can be added to the right of the $\displaystyle \tanh^{-1}$ equation and the corresponding $\displaystyle \tan$ terms multiplied to the right of the $\displaystyle \tan$ equation.