7. Given that
$$I _ { n } = \int \frac { \sin n x } { \sin x } \mathrm {~d} x , \quad n \geqslant 1$$
- prove that, for \(n \geqslant 3\)
$$I _ { n } - I _ { n - 2 } = \int 2 \cos ( n - 1 ) x \mathrm {~d} x$$
- Hence, showing each step of your working, find the exact value of
$$\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 6 } } \frac { \sin 5 x } { \sin x } d x$$
giving your answer in the form \(\frac { 1 } { 12 } ( a \pi + b \sqrt { 3 } + c )\), where \(a\), \(b\) and \(c\) are integers to be found.