# Prove $x^2+\frac{1}{x^2}=2\cos(2\theta)$

Prove $$x^2+\frac{1}{x^2}=2\cos(2\theta)$$ and $$x^3+\frac{1}{x^3}=2\cos(3\theta)$$ knowing that there exist a number $$x$$ given angle $$\theta$$ such that $$x+\frac{1}{x}=2\cos(\theta)$$

Doesn't really know how to start this problem, thought that I would some how need to use the double angle identities

• The number $x$ is necessarily real? – José Carlos Santos Dec 12 '19 at 9:26
• Have you tried squaring both sides? – T.J. Gaffney Dec 12 '19 at 9:28

$$\left(x+\frac 1x\right)^2 = x^2+2+\frac 1{x^2} = 4\cos^2\theta \\\implies x^2+\frac{1}{x^2} = 2(2\cos^2\theta-1) = 2\cos2\theta$$

Similarly

$$\left(x+\frac 1x\right)^3 = x^3 + 3\left(x+\frac 1x\right) + \frac{1}{x^3} = x^3+\frac{1}{x^3} + 6\cos\theta = 8\cos^3\theta$$

$$\implies x^3+\frac{1}{x^3} = 2(4\cos^3\theta-3\cos\theta) = 2\cos3\theta$$

• My sincere congratulations. – Sebastiano Dec 12 '19 at 9:41
• @Sebastiano Thank you! – Ak. Dec 12 '19 at 9:42
• +1 for a beautiful answer! – Toby Mak Dec 12 '19 at 9:45
• Thank you!! @TobyMak – Ak. Dec 12 '19 at 9:53
• Nice and elegant (+1) – trancelocation Dec 12 '19 at 11:57

$$x=\cos\theta\pm i\sin\theta$$

Taking $$+$$ sign for integer $$n$$,

using https://en.m.wikipedia.org/wiki/De_Moivre's_formula,

$$x^n=\cos n\theta+i\sin n\theta$$

$$1/x^n=?$$

Similarly consider $$-$$ sign