Question 5 - A-Level Maths

Express $\tan 2 \alpha$ in terms of $\tan \alpha$ and hence solve, for $0 ^ { \circ } < \alpha < 180 ^ { \circ }$, the equation $$\tan 2 \alpha \tan \alpha = 8 .$$
Given that $\beta$ is the acute angle such that $\sin \beta = \frac { 6 } { 7 }$, find the exact value of
1. $\operatorname { cosec } \beta$,
2. $\cot ^ { 2 } \beta$.