Given that $\sin x + \cos x = \frac{2}{3}$, find $\cos 4x$.

${\sin ^2}x + {\cos ^2}x + 2\sin x\cos x = \frac{4}{9}$     (M1)(A1)

using ${\sin ^2}x + {\cos ^2}x = 1$     (M1)

$2\sin x\cos x =  - \frac{5}{9}$

using $2\sin x\cos x = \sin 2x$     (M1)

$\sin 2x =  - \frac{5}{9}$

$\cos 4x = 1 - 2{\sin ^2}2x$     M1

Note:     Award this M1 for decomposition of cos 4x using double angle formula anywhere in the solution.

$= 1 - 2 \times \frac{{25}}{{81}}$

$= \frac{{31}}{{81}}$     A1

[6 marks]