a) Use correct $\cos \left( {A + B} \right)$ formula to express $\cos 3\theta $ in terms of trig functions of $2\theta $ and $\theta $

Use correct trig formulae and Pythagoras to express $\cos 3\theta {\text{ }}$ in terms of $\cos \theta $

Obtain a correct expression in terms of $\cos \theta $ in any form

Obtain the given identity correctly

[SR: Give M1 for using correct formulae to express RHS in terms of $\cos \theta $ and $\cos 2\theta $, then M1A1 for expressing in terms of either only $\cos 3\theta $ and $\cos \theta $, or only $\cos 2\theta $, $\sin 2\theta $, $\cos \theta $, and $\sin \theta $, and A1 for obtaining the given identity correctly.]

b) Use identity and integrate, obtaining terms $\frac{1}{4}(\frac{1}{3}\sin 3\theta )$ and $\frac{1}{4}(3\sin \theta )$, or equivalent

Use limits correctly in an integral of the form $k\sin 3\theta  + l\sin \theta $

Obtain answer $\frac{2}{3} - \frac{3}{8}\sqrt 3 $, or any exact equivalent