Almost pizza

Note first that $\tan(x) = \cot(\frac{\pi}{2} - x)$ , and so $\tan(\frac{\pi z}{2}) = \cot(\frac{\pi}{2}(1 - z))$ .

Now let $y = 1 - z$ . Then $y \to 0$ as $z \to 1$ , and the limit then becomes

$\displaystyle\lim_{y \to 0} y*\cot(\frac{\pi}{2}y) = \dfrac{2}{\pi} \lim_{y \to 0}\dfrac{\frac{\pi}{2}y}{\tan(\frac{\pi}{2}y)} = \dfrac{2}{\pi}$ ,

since $\displaystyle\lim_{x \to 0} \dfrac{\tan(x)}{x} = 1$ . Thus $a = \boxed{2}$ .

Observing graph, we see that the answer is $\Large \frac {2}{\pi}$ . So, $a$ = $2$ . $_\square$