Evaluating $16\cos^6\left(\frac\pi{18}\right) - 24\cos^4\left(\frac{\pi}{18}\right) + 9\cos^2\left(\frac\pi{18}\right)$

Question

What’s the value of the given expression?

$$16\cos^6\left(\frac\pi{18}\right) - 24\cos^4\left(\frac{\pi}{18}\right) + 9\cos^2\left(\frac\pi{18}\right)$$

Thanks in advance.

Answer 1

$$ 16\cos^6\left(\frac\pi{18}\right) - 24\cos^4\left(\frac{\pi}{18}\right) + 9\cos^2\left(\frac\pi{18}\right)= $$ $$ =\left[4\cos^3\left(\frac\pi{18}\right)-3\cos\left(\frac\pi{18}\right)\right]^2=\left[\cos\left(3\frac{\pi}{18} \right) \right]^2=... $$

Answer 2

Since the minimal polynomial of $\cos\left(\frac{\pi}{18}\right)$ over $\mathbb{Q}$ is $64x^6-96x^4+36x^2-3$ the outcome is $\color{red}{\frac{3}{4}}$.