Poles of $\tan z$ at odd multiples of $\pi/2$ and their residues

Prove that

\[ \mathrm { ( a ) } \operatorname* { l i m } _ { z \to ( 2 k + 1 ) \pi / 2 } \tan z = \infty , \mathrm { ( b ) } \operatorname* { l i m } _ { z \to ( 2 k + 1 ) \pi / 2 } [ z - ( 2 k + 1 ) \pi / 2 ] \tan z = - 1 . \]

[Hint: The proof of (b) is very short if done the “right” way.]