Polynomials with Trigonometric Solutions

(a) Solve $4x^3-3x=\frac{1}{2}$. (b) Solve $x^3-3x-1=0$. (c) Use the cosine and inverse cosine functions to devise a general formula for the solution of $x^3+px+q=0$ by the method outlined in these problems. Under what conditions can $x^3+px+q=0$ be solved by this method?

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(a) Use de Moivre’s theorem to prove the identity

$$\cos 6\alpha =32{\cos }^6\alpha -48{\cos }^4\alpha +18{\cos }^2\alpha -1$$

(b) Hence show that the polynomial equation $32x^6-48x^4+18x^2-1=0$ has roots $x=\cos \frac{n\pi }{12}$, for $n=1,3,5,7,9,11$.

(c) Use the product of these six roots to deduce that $\cos \frac{\pi }{12}\cos \frac{5\pi }{12}=\frac{1}{4}$.