This can be expanded:
cosxtany+cosxcosy-sinxsiny+
sinxsec²xdx/dx+cosxcosydy/dx-sinxsinydy/dx=0
Now rearrange the terms:
(cosxtany+sinxsec²ydy/dx)+
(-sinxsiny+cosxcosydy/dx)+
(cosxcosy-sinxsinydy/dx)=0.
Each of these pairs is a derivative:
d(sinxtany)/dx+d(cosxsiny)/dx+d(sinxcosy)/dx=0.
Now we can integrate:
sinxtany+cosxsiny+sinxcosy=C, constant of integration.
This can also be written: sinxtany+sin(x+y)=C.