Use the compound angle identity to show that \(\sin\dfrac\pi{12}=\dfrac{\sqrt6-\sqrt2}4\).[3]
Show that, for any \(m\in\mathbb Z^+\), \[2\sin\theta\cos[(m+1)\theta]+\sin m\theta=\sin[(m+2)\theta].\][4]
Use Mathematical Induction to prove \[\sum_{r=1}^n\cos[(2r-1)\theta]=\frac{\sin2n\theta}{2\sin\theta}\quad\text{for all}\hspace{0.5em}n\in\mathbb Z^+.\][6]