Steps or techniques to prove

Question

I agree that I have my senior secondary homework assignment to do but I am unable to understand how to proceed on the below question:

If $2 \cos \theta = x + 1/x$ show that $\cos 2 \theta = \frac{1}{2}(x^2 + 1/x^2)$.

How can I prove the required?

Answer 1

Hint: $$\cos(2\theta)=2\cos^2(\theta)-1.$$

Answer 2

Another way:

$$x^2-2x\cos\theta+1=0$$

$$x=\cos\theta\pm i\sin\theta=e^{\pm i\theta}$$ using How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?

$$x^n=e^{\pm in\theta}\implies x^{-n}=e^{-\mp in\theta}$$

$$x^n+x^{-n}=2\cos n\theta$$ where $n$ is any integer