Why is it that $\tan\left(\frac{183.5^\circ}{2}\right)=-32.73\ldots$, but $2\arctan(-32.73)=-176.5^\circ$?

Question

This seems very strange to me that when on calculator I write $$\tan\left(\frac{183.5^\circ}{2}\right)$$ the calculator gives me the answer $-32.73026372$ but when I write $$ 2\arctan(-32.73.73026372)$$ the calculator gives me the answer $-176.5$.

So I wonder why does this inconsistency lie in mathematically sound procedures?

Answer 1

This is not a contradiction because nobody is claiming that $\arctan$ is the inverse function of $\tan$. $\tan$ is not a one-to-one function, so there are values $x$ and $y$ such that $x\neq y$ but $\tan(x)=\tan(y)$.

Answer 2

As Matt Samuel correctly answered, $\tan$ is not a one-to-one function. In particular,

$\tan (183.5^\circ/2)=\tan(91.75^\circ)=\tan(91.75^\circ-180^\circ)=\tan(-88.25^\circ)=\tan(-176.5^\circ/2).$