`y=sin(x+y)`

Prove that |(1,x,y),(0, sin x, sin y),(0, cos x,cos y)|= sin (x-y)

Find dy/dx when (sin x)^y = (sin y)^x

[x (cos y) / (x) + y (sin y) / (x)] y = [y (sin y) / (x) -x (cos y) / (x)] x ((dy) / (dx))

{ xcos ((y)/(x)) + y sin ((y)/(x))} y dx = { y sin ((y)/(x)) - x cos ((y)/(x))}x dy

{ xcos ((y)/(x)) + y sin ((y)/(x))} y dx = { y sin ((y)/(x)) - x cos ((y)/(x))}x dy

Solve {x cos((y)/(x))+y sin((y)/(x))}ydx={y sin((y)/(x))-x cos((y)/(x))}xdy

{x cos (y/x)+y sin (y/x)} y d x={y sin (y/x)-(x) cos (y/x)} x d y

The solution of differential equation [x cos ((y) / (x)) + y sin ((y) / (x))] y- [y sin ((y) / (x)) - x cos ((y ) / (x))] x (dy) / (dx) = 0