Product of a sine and a cosine of shifted angles expressed in terms of cos 2x, an exact value and an equation

By expanding \(\sin\left(x+\frac{\pi}{3}\right)\) and \(\cos\left(x+\frac{\pi}{6}\right)\) with the compound-angle formulae, show that \(4\sin\left(x+\frac{\pi}{3}\right)\cos\left(x+\frac{\pi}{6}\right)=1+2\cos2x\).[3]

Deduce the exact value of \(4\sin\frac{3\pi}{4}\cos\frac{7\pi}{12}\).[3]

Find all the solutions of \(4\sin\left(x+\frac{\pi}{3}\right)\cos\left(x+\frac{\pi}{6}\right)=1+\sqrt3\) in the interval \(0\le x\le2\pi\).[3]