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!
If it has been asked reversely
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Ks/kt=cp/cv
Great! No Maxwell relations needed!
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