There are some well-known exact values for trig functions, such as $$\sin\frac{\pi}{6}=\frac{1}{2},\quad \tan\frac{\pi}{3}=\sqrt 3, \quad\text{etc.}$$ Are there comparable special values for the hyperbolic trig functions?
The output should be expressible as sums, differences, products, quotients and $n$-th roots of integers. This paper gives some examples of what I am talking about.
If $x^2-dy^2=1$ say, then $$ \sinh(\log(x+\sqrt{d}y))=\frac{1}{2}\left(x+\sqrt{d}y-\frac{1}{x+\sqrt{d}y}\right)=\frac{1}{2}\left(x+\sqrt{d}y-(x-\sqrt{d}y)\right)=\sqrt{d}y, $$ $$ \cosh(\log(x+\sqrt{d}y))=\frac{1}{2}\left(x+\sqrt{d}y+\frac{1}{x+\sqrt{d}y}\right)=\frac{1}{2}\left(x+\sqrt{d}y+x-\sqrt{d}y\right)=x, $$ and similarly with other integral Pellian equations.