Please help me how to deal with maximization of function
$$f(x,y)=1-e^{-\pi x}+e^{\pi x}\left[1-\cos(\pi y)+\sin(\pi y)\right]$$
subject to the constraint $x^2+y^2=R^2$.
Using Lagrange multipliers one has: $$\pi e^{\pi x}\left[\cos(\pi y)+\sin(\pi y)\right]-2\lambda y=0$$ $$\pi \left[e^{-\pi x}+e^{\pi x}\left(1-\cos(\pi y)+\sin(\pi y)\right)\right]-2\lambda x=0$$ $$x^2+y^2=R^2$$ but this system is too hard for me.
Any suggestions please?
Is $x=\frac{1}{2\pi}\ln\left(\frac{1}{\sqrt 2 -1}\right), y=-\frac{1}{4}, \lambda=0$ the only solution of this system?