Thermodynamics proving cp/cv = kt/ks

[Chronos000]

Homework Statement

at some point in my derivation I get to the following line:

cp/cv = (dS/dP)_T(dP/dT)_S / (dS/dV)_T(dV/dT)_S

I think this equals (dS/dP)_T(dS/dV)_P / (dP/dT)_V(dV/dT)_S

- from applying maxwells relations.
but my book says it is in fact the following:

(dS/dP)_T(dV/dS)_T / (dV/dT)_S(dT/dP)_S

I haven't a clue how they are doing this

 

[Mapes]

Aren't they just using $1/(dX/dY)_Z=(dY/dX)_Z$?