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if x/y=1 then x=y. y=x. a straight line.

But if xArctan(x^y)=ysin(x) and y,x are not constants (they are variables), then you can find the rate of change of y with respect to x. So:
x(arctan(x^y))=ysin(x), in this, i Will write dy/dx 'f'.
ycos(x)+(f)sin(x)=arctan(x^y)+(x)(derivative of arctan(x^y)).
Now differentiating arctan(x^y) with respect to x where y is variable: (derivative of x^y)/(1+(x^2y)).
derivative of x^y: let a=x^y, then ln(a)=yln(x), now it is easy. da/dx=(x^y)(yx^-1 +(f)ln(x))
so d/dx (arctan(x^y)) gives yx^y-1 +(f)(x^y)ln(x) all divided by 1+x^2y
Substituting it into original equation:
ycos(x)+fsin(x)=arctan(x^y)+x(yx^y-1 +(f)(x^y)ln(x))/(1+x^2y), now its easy, just rearrange to make f the subject.

f= ((1+x^2y)(ycos(x)-arctan(x^y)-yx^y)/((x^y)(ln(x))-(1+x^2y)(sin(x))