Proving that Cp/Cv=Kt/Ks

Here is the thermodynamic proof that Cp/Cv=Kt/Ks.  It is pretty straight forward if you use the cyclic chain rule.

I hope this helps some of you. I know this is a common question that is often asked in Physical Chemistry (P-Chem). If this is helpful and you would like to see the proof for another thermodynamic identity, leave me a comment and let me know!