Anyway here is my proof.
cot T - tan T = 2 cot 2T
iff cos T / sin T - sin T / cos T = 2 cos 2T / sin 2 T
iff (cos^2 T - sin^2 T) / sin T cos T = 2 (cos^2 T - sin^T) / 2 sin T cos T
QED.
Identities used: cos 2T = cos^2 T - sin^2 T and sin 2T = 2 sin T cos T