The value of $\underset{x→1}{\lim}(1-x)\tan(\frac{πx}{2})$ is:

$\frac{2}{π}$

$\underset{x→1}{\lim}(1-x)\tan(\frac{πx}{2})=\underset{h→0}{\lim}(-h)\tan(\frac{π}{2}(1+h))$

$\underset{h→0}{\lim}(+h)\cot(\frac{π}{2}h)=+\underset{h→0}{\lim}\frac{h}{\tan\frac{π}{2}h}=\frac{2}{π}\underset{h→0}{\lim}\frac{\frac{π}{2}h}{\tan\frac{π}{2}h}=\frac{2}{π}$