This was a very surprising discovery for me that identities like this exist:

$$\tan \frac{c}{2}=\tan \frac{a}{2}\tan \frac{b}{2} \qquad \rightarrow$$

$$\tanh^{-1} (\cos a)+\tanh^{-1} (\cos b)=\tanh^{-1} (\cos c)$$

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):

$$\tan \frac{c}{2}=\frac{\tan \frac{a}{2}-\tan \frac{b}{2}}{\tan \frac{a}{2}+\tan \frac{b}{2}} \qquad \rightarrow$$

$$\tanh^{-1} (\cos b)-\tanh^{-1} (\cos a)=\tanh^{-1} (\sin c)$$

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?

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?