7 The equation of a curve is \(y = \frac { x } { \cos ^ { 2 } x }\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\). At the point where \(x = a\), the tangent to the curve has gradient equal to 12 .
- Show that \(a = \cos ^ { - 1 } \left( \sqrt [ 3 ] { \frac { \cos a + 2 a \sin a } { 12 } } \right)\).
- Verify by calculation that \(a\) lies between 0.9 and 1 .
- Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.