Prove that \[\cos 3\theta \sin 2\theta - \cos 4\theta \sin \theta = \cos 2\theta \sin \theta.\]

- Starting from the relations
\[\begin{align*}
\sin(A+B) &{}= \sin A \cos B + \cos A \sin B,\\
\cos(A+B) &{}= \cos A \cos B - \sin A \sin B,
\end{align*}\]
obtain an expression for \(\tan(A+B)\) in terms of \(\tan A\) and \(\tan B\).

Without the use of a calculator or tables, show that the sum of the three acute angles whose tangents are \(\dfrac{2}{9}\), \(\dfrac{1}{4}\) and \(\dfrac{1}{3}\) is \(\dfrac{\pi}{4}\).