If $x$ and $y$ are real and $x^2 + y^2 = 1$, compute the maximum value of $(x+y)^2.$ If $x$ and $y$ are real and $x^2 + y^2 = 1$, compute the maximum value of $(x+y)^2.$

Should I manipulate the equation?  I'm stuck on this problem.  Answers are greatly appreciated.

Trig-free solution: If $x^2+y^2=1$ then
$$
(x+y)^2 + (x-y)^2 = 2(x^2+y^2)=2
$$
so the largest value of $(x+y)^2$ is $2$, attained when $x=y$.
