Question 4 - A-Level Maths

Prove that \(\frac { 1 - \cos 2 \theta } { 1 + \cos 2 \theta } \equiv \tan ^ { 2 } \theta\).
Hence find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { 1 - \cos 2 \theta } { 1 + \cos 2 \theta } \mathrm {~d} \theta\).