If x satisfies the equation \(\sin \left( {x + \frac{\pi }{3}} \right) = 2\sin x\sin \left( {\frac{\pi }{3}} \right)\), show that \(11\tan x = a + b\sqrt 3 \), where a, b \( \in {\mathbb{Z}^ + }\).

\(\sin \left( {x + \frac{\pi }{3}} \right) = \sin x\cos \left( {\frac{\pi }{3}} \right) + \cos x\sin \left( {\frac{\pi }{3}} \right)\)     (M1)

\(\sin x\cos \left( {\frac{\pi }{3}} \right) + \cos x\sin \left( {\frac{\pi }{3}} \right) = 2\sin x\sin \left( {\frac{\pi }{3}} \right)\)

\(\frac{1}{2}\sin x + \frac{{\sqrt 3 }}{2}\cos x = 2 \times \frac{{\sqrt 3 }}{2}\sin x\)     A1

dividing by \(\cos x\) and rearranging     M1

\(\tan x = \frac{{\sqrt 3 }}{{2\sqrt 3 - 1}}\)     A1

rationalizing the denominator     M1

\(11\tan x = 6 + \sqrt 3 \)     A1

[6 marks]

Most candidates were able to make a meaningful start to this question, but a significant number were unable to find an appropriate expression for \(\tan x\) or to rationalise the denominator.