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]