a) Express $\cos 4\theta $ as $2{\cos ^2}{\text{ }}2\theta  - 1$ or ${\cos ^2}2\theta  - {\sin ^2}{\text{ }}2\theta $ or $1 - 2{\sin ^2}2\theta $

Express $\cos 4\theta $ in terms of $\cos \theta $

Obtain $8{\text{ }}{\cos ^4}\theta  - 8{\text{ }}{\cos ^2}\theta  + 1$

Use $\cos 2\theta  = 2{\text{ }}{\cos ^2}\theta  - 1$ to obtain given answer $8{\text{ }}{\cos ^4}\theta  - 3$

b)(i) State or imply ${\cos ^4}\theta  = \frac{1}{2}$

Obtain $0.572$

Obtain $ - 0.572$

(ii) Integrate and obtain form ${k_1}\theta  + {k_2}\sin 4\theta  + {k_3}\sin 2\theta $

Obtain $\frac{3}{8}\theta  + \frac{1}{{32}}\sin 4\theta  + \frac{1}{4}\sin 2\theta $

Obtain $\frac{3}{{32}}\pi  + \frac{1}{4}$ following completely correct work