Answer
See explanation
Work Step by Step
Using the definition of $c_v$, $$
c_v=T\left(\frac{\partial s}{\partial T}\right)_v=T\left(\frac{\partial s}{\partial P}\right)_v\left(\frac{\partial P}{\partial T}\right)_v
$$ Substituting the first Maxwell relation $\left(\frac{\partial S}{\partial P}\right)_v=-\left(\frac{\partial v}{\partial T}\right)_x$,
$$ c_v=-T\left(\frac{\partial v}{\partial T}\right)_s\left(\frac{\partial P}{\partial T}\right)_v
$$ Using the definition of $c_p$, $$
c_p=T\left(\frac{\partial s}{\partial T}\right)_P=T\left(\frac{\partial s}{\partial v}\right)_P\left(\frac{\partial v}{\partial T}\right)_P
$$ Substituting the second Maxwell relation $\left(\frac{\partial s}{\partial v}\right)_P=\left(\frac{\partial P}{\partial T}\right)_s$,
$$
c_P=T\left(\frac{\partial P}{\partial T}\right)_s\left(\frac{\partial v}{\partial T}\right)_P
$$
