Lagrange multipliers are basically used to find the maximum and minimum points (or critical points) of a function given that a constraint function is defined. I have been trying all sorts of ways to solve the following problem and I just could not find a way to determine critical points. The question is:
Constraint function $$x+y+\arccos(-\cos(x+y))=\pi$$
Function $$f(x,y)=\frac{1}{2}\sin(x+y)$$
As you can see, since these are trigonometric function - there partial derivatives $f_{x}$ and $f_{y}$ are same which becomes an issue because then lambda (multiplier) in both cases are exactly the same.
One other thing to note is that $x$ and $y$ are angles and their ranges go from $0$ to $\pi/2$. There definitely exists a maximum because the function above represents an area of a triangle inscribed in a circle and from my own experimentation I believe that the maximum area (or function) can go near 0.5.
If you feel that this question has been asked before, please give me a link to the question and if any further information needs to be provided please let me know. Thank you in advance.
EDIT
I have found the solution I was looking for but I am still open to other derivations of solutions. If anyone wants to see how I found the answer feel free to ask and I will post it in here.