The composition of circular trigonometric functions, like $\cos(\tan^{-1}(x))$, can be derived drawing a right angle triangle and applying Pythagoras' theorem and the definition of sine and cosine in terms of the sides of the triangle. This is shown for example in this wikipedia page: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Relationships_among_the_inverse_trigonometric_functions
I tried to find an analog for hyperbolic trig functions, but I haven't found anything. Does there exist a similar mnemonic for them?
The hyperbolic equivalent to the trigonometric mnemonic $SOHCAHTOA$ is $$CHASOATOH$$
If you are familiar with $SOHCAHTOA$ then the hyperbolic one should be self-explanatory.