On two functional equations and their solutions

The present work aims to determine the solution f:R2→Rf:R2→R of the equation f(ux−vy,uy−vx)=f(x,y)+f(u,v)+f(x,y)f(u,v) for all x,y,u,v∈Rx,y,u,v∈R without any regularity assumption. The solution of the functional equation f(ux+vy,uy−vx)=f(x,y)+f(u,v)+f(x,y)f(u,v) is also determined. The methods of solution of these equations are simple and elementary. These two equations arise in connection with the characterizations of determinant and permanent of two-by-two symmetric matrices, respectively.