interesting scalene triangle trigonometry problem the problem is to find the minimum value of cos(2α)+cos(2β)+cos(2γ) in a scalene triangle. So what should the value of α,β, and γ be?
I already transformed the expression using γ=180∘−(α+β) to cos(2α)+cos(2β)+cos(2α+2β)
so how can I finish the problem if this is a good way?
Thank you

Given $A+B+C = \pi$ and $$\cos 2A+\cos 2 B+\cos 2C = 2\cos(A+B)\cos(A-B)+\cos 2C$$
So $$\cos 2A+\cos 2B+\cos 2C = -2\cos C\cdot \cos(A-B)+2\cos^2 C-1$$
$$ = -2\cos C\left[\cos(A-B)-\cos(C)\right]-1=-2\cos C\left[\cos(A-B)+\cos(A-B)\right]$$
$$ = -4\cos A \cos B\cos C-1$$
So $$\cos 2A+\cos 2B+\cos 2C = -1-4\cos A\cos B\cos C\geq -1+\frac{4}{8}=-\frac{3}{2}$$
Above we have used $$\cos A\cos B\cos C\leq \frac{1}{8}$$
and equality hold when $\displaystyle A=B=C=\frac{\pi}{3}$
