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Question: Why does `solve` miss the solutions without warnings?

Question:Why does `solve` miss the solutions without warnings?

sursumCorda 1169 Maple 2024
solve
December 11 2024
2

The output RealDomain:-solve(x**2 + 2*cos(x) = (Pi/3)**2 + 1, [x]) means that there is no real solution, but clearly, both x = -Pi/3 and x = +Pi/3 satisfy the original equation: 

So, why does `solve` lose the real solutions without any warning messages? 
Code: 

restart;
eq := 9*(x^2 + 2*cos(x)) = Pi^2 + 9;
RealDomain:-solve(eq, [x]);
                               []

:-solve({eq, x >= 0}, [x]); # as (lhs - rhs)(eq) is an even function 
                               []
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