11 Prove de Moivre's theorem for a positive integral exponent: $$\text { for all positive integers } n , \quad ( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta \text {. }$$ Use de Moivre's theorem to show that $$\cos 7 \theta = 64 \cos ^ { 7 } \theta - 112 \cos ^ { 5 } \theta + 56 \cos ^ { 3 } \theta - 7 \cos \theta$$ Hence obtain the roots of the equation $$128 x ^ { 7 } - 224 x ^ { 5 } + 112 x ^ { 3 } - 14 x + 1 = 0$$ in the form \(\cos q \pi\), where \(q\) is a rational number.