Question

Evaluate: 8 cos cube ( pie /9)-6 cos( pie /9)?

Answer 1

$$\begin{gathered} Given, \\ 8{\cos }^3\left(\frac{\mathrm{\pi }}{9}\right)-6\cos \left(\frac{\mathrm{\pi }}{9}\right) \\ =2\left(4{\cos }^3\left(\frac{\mathrm{\pi }}{9}\right)-3\cos \left(\frac{\mathrm{\pi }}{9}\right)\right) \\ =2\cos \left(3\times \frac{\mathrm{\pi }}{9}\right)\left[Since\cos 3x=4{\cos }^{3}x-3\cos x\right] \\ =2\cos \left(\frac{\mathrm{\pi }}{3}\right) \\ =2\times \frac{1}{2} \\ =1 \end{gathered}$$