Answer: (a) \(\operatorname{cosec}7\theta=\dfrac{\operatorname{cosec}^{7}\theta}{7\operatorname{cosec}^{6}\theta-56\operatorname{cosec}^{4}\theta+112\operatorname{cosec}^{2}\theta-64}\).
(b) The roots are \(x=\operatorname{cosec}(q\pi)\), where
\(q=\dfrac{1}{42},\dfrac{5}{42},\dfrac{13}{42},\dfrac{17}{42},-\dfrac{7}{42},-\dfrac{11}{42},-\dfrac{19}{42}\).
(a) Let \(c=\cos\theta\) and \(s=\sin\theta\). By de Moivre's theorem,
\((c+is)^7=\cos 7\theta+i\sin 7\theta\),
so \(\sin 7\theta\) is the imaginary part of \((c+is)^7\):
\(\sin 7\theta=7c^6s-35c^4s^3+21c^2s^5-s^7.\)
Using \(c^2=1-s^2\),
\(\sin 7\theta=7(1-s^2)^3s-35(1-s^2)^2s^3+21(1-s^2)s^5-s^7.\)
Now expand:
\((1-s^2)^2=1-2s^2+s^4\),
\((1-s^2)^3=1-3s^2+3s^4-s^6\).
Hence
\(\sin 7\theta=7(1-3s^2+3s^4-s^6)s-35(1-2s^2+s^4)s^3+21(1-s^2)s^5-s^7,\)
which simplifies to
\(\sin 7\theta=7s-56s^3+112s^5-64s^7.\)
Therefore
\(\operatorname{cosec}7\theta=\frac{1}{7s-56s^3+112s^5-64s^7}.\)
Divide numerator and denominator by \(s^7\):
\(\operatorname{cosec}7\theta=\frac{s^{-7}}{7s^{-6}-56s^{-4}+112s^{-2}-64}.\)
Since \(s^{-1}=\operatorname{cosec}\theta\),
\(\operatorname{cosec}7\theta=\frac{\operatorname{cosec}^{7}\theta}{7\operatorname{cosec}^{6}\theta-56\operatorname{cosec}^{4}\theta+112\operatorname{cosec}^{2}\theta-64}.\)
(b) Rewrite the equation as
\(x^7-14x^6+112x^4-224x^2+128=0\)
\(\Rightarrow x^7=14x^6-112x^4+224x^2-128=2(7x^6-56x^4+112x^2-64).\)
So
\(\frac{x^7}{7x^6-56x^4+112x^2-64}=2.\)
From part (a), if \(x=\operatorname{cosec}\theta\), then
\(\frac{x^7}{7x^6-56x^4+112x^2-64}=\operatorname{cosec}7\theta.\)
Hence
\(\operatorname{cosec}7\theta=2,\)
so
\(\sin 7\theta=\frac12.\)
Thus
\(7\theta=\frac\pi6+2k\pi \quad \text{or} \quad 7\theta=\frac{5\pi}6+2k\pi.\)
Therefore
\(\theta=\frac\pi{42}+\frac{2k\pi}{7} \quad \text{or} \quad \theta=\frac{5\pi}{42}+\frac{2k\pi}{7}.\)
Taking the seven distinct values modulo \(2\pi\), we obtain
\(\theta=\frac\pi{42},\ \frac{5\pi}{42},\ \frac{13\pi}{42},\ \frac{17\pi}{42},\ -\frac{7\pi}{42},\ -\frac{11\pi}{42},\ -\frac{19\pi}{42}.\)
Hence the seven roots are
\(x=\operatorname{cosec}\left(\frac\pi{42}\right),\ \operatorname{cosec}\left(\frac{5\pi}{42}\right),\ \operatorname{cosec}\left(\frac{13\pi}{42}\right),\ \operatorname{cosec}\left(\frac{17\pi}{42}\right),\ \operatorname{cosec}\left(-\frac{7\pi}{42}\right),\ \operatorname{cosec}\left(-\frac{11\pi}{42}\right),\ \operatorname{cosec}\left(-\frac{19\pi}{42}\right).\)
