<s>[INST] Prove that $\cos^2(\theta) + \cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})=3/2$ 
Prove that $$\cos^2(\theta) + \cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})=\frac{3}{2}$$

I thought of rewriting $$\cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})$$ as $$\cos^2(90^{\circ}+ (\theta +30^{\circ})) + \cos^2(90^{\circ}-(\theta-30^{\circ}))$$ However I don't seem to get anywhere with this.
Unfortunately I don't know how to solve this question. I would be really grateful for any help or suggestions. Many thanks in advance!
 [/INST] $\cos^2(A-120^\circ)+\cos^2(A+120^\circ)=1+\cos^2(A-120^\circ)-\sin^2(A+120^\circ)$
and Using Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$, $\cos^2(A-120^\circ)-\sin^2(A+120^\circ)=1+\cos(2A)\cos(240^\circ)=?$
$\cos(240^\circ)=\cos(180^\circ+60^\circ)=-\cos60^\circ=?$
Now use $\cos2A=2\cos^2A-1$
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