Differentiate both sides with respect to x, using the chain rule for the sine term:
[tex]\cos(x) + \sin(y) = 1[/tex]
[tex]\implies -\sin(x) + \cos(y)\dfrac{\mathrm dy}{\mathrm dx} = 0[/tex]
Solve for dy/dx :
[tex]-\sin(x) + \cos(y)\dfrac{\mathrm dy}{\mathrm dx} = 0 \\\\ \cos(y) \dfrac{\mathrm dy}{\mathrm dx} = \sin(x) \\\\ \boxed{\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{\sin(x)}{\cos(y)}}[/tex]
