Answer
$$\eqalign{
& {z_s} = \cos \left( {s + t} \right)\cos 2\left( {s - t} \right) - 2\sin \left( {s + t} \right)\sin 2\left( {s - t} \right) \cr
& {z_t} = \cos \left( {s + t} \right)\cos 2\left( {s - t} \right) + 2\sin \left( {s + t} \right)\sin 2\left( {s - t} \right) \cr} $$
Work Step by Step
$$\eqalign{
& {\text{Let the functions }}z = \sin x\cos 2y,\,\,\,\,\,x = s + t{\text{ and }}y = s - t \cr
& \cr
& {\text{Calculate }}{z_s},{\text{ apply }}\frac{{\partial z}}{{\partial s}} = \frac{{\partial z}}{{\partial x}}\frac{{\partial x}}{{\partial s}} + \frac{{\partial z}}{{\partial y}}\frac{{\partial y}}{{\partial s}} \cr
& \frac{{\partial z}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {\sin x\cos 2y} \right] = \cos x\cos 2y \cr
& \frac{{\partial z}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {\sin x\cos 2y} \right] = - 2\sin x\sin 2y \cr
& \frac{{\partial x}}{{\partial s}} = \frac{\partial }{{\partial s}}\left[ {s + t} \right] = 1 \cr
& \frac{{\partial y}}{{\partial s}} = \frac{\partial }{{\partial s}}\left[ {s - t} \right] = 1 \cr
& {\text{Then}}{\text{,}} \cr
& \frac{{\partial z}}{{\partial s}} = \left( {\cos x\cos 2y} \right)\left( 1 \right) + \left( { - 2\sin x\sin 2y} \right)\left( 1 \right) \cr
& \frac{{\partial z}}{{\partial s}} = \cos x\cos 2y - 2\sin x\sin 2y \cr
& {\text{Write in terms of }}s{\text{ and }}t \cr
& \frac{{\partial z}}{{\partial s}} = \cos \left( {s + t} \right)\cos 2\left( {s - t} \right) - 2\sin \left( {s + t} \right)\sin 2\left( {s - t} \right) \cr
& \cr
& {\text{Calculate }}{z_t},{\text{ apply }}\frac{{\partial z}}{{\partial t}} = \frac{{\partial z}}{{\partial x}}\frac{{\partial x}}{{\partial t}} + \frac{{\partial z}}{{\partial y}}\frac{{\partial y}}{{\partial t}} \cr
& \frac{{\partial x}}{{\partial t}} = \frac{\partial }{{\partial t}}\left[ {s + t} \right] = 1 \cr
& \frac{{\partial y}}{{\partial t}} = \frac{\partial }{{\partial t}}\left[ {s - t} \right] = - 1 \cr
& {\text{Then}}{\text{,}} \cr
& \frac{{\partial z}}{{\partial t}} = \left( {\cos x\cos 2y} \right)\left( 1 \right) + \left( { - 2\sin x\sin 2y} \right)\left( { - 1} \right) \cr
& \frac{{\partial z}}{{\partial t}} = \cos x\cos 2y + 2\sin x\sin 2y \cr
& {\text{Write in terms of }}s{\text{ and }}t \cr
& \frac{{\partial z}}{{\partial t}} = \cos \left( {s + t} \right)\cos 2\left( {s - t} \right) + 2\sin \left( {s + t} \right)\sin 2\left( {s - t} \right) \cr} $$
