Give the proof of the inequality $0.5(|A+B|+|A-B|)= \max (|A|,|B|)$ $$0.5(|A+B|+|A-B|)= \max (|A|,|B|)$$
where $A$ and $B$ are any values (real number)
I have tried to solve this using triangle inequality but got $0.5(|A+B|+|A-B|)> |A|$
How to prove the given inequality??

WLOG, $A\ge B\ge0$ because swapping $A,B$ and/or changing one or two signs has no effect.
Then the identity reduces to
$$\frac{A+B+A-B}2=A,$$ which is true.
