<s>[INST] Proving $\cos a-\cos b-\cos c\geq -\frac{3}{2}$, where $a+b+c=2\pi$ and $a,b,c>0$ I need help proving the following trig inequality:

$$\cos a-\cos b-\cos c\geq -\frac{3}{2}$$ where $a+b+c=2\pi$ and $a,b,c>0$.

I've found that equality occurs when $a=\frac{4\pi}{3},b=c=\frac{\pi}{3},$ but I'm not sure how to prove it in general.
I would also like to only use math covered in precalc and below, if that matters.
 [/INST] We need to prove that $$\cos{a}-2\cos\frac{b+c}{2}\cos\frac{b-c}{2}+\frac{3}{2}\geq0$$ or
$$2\cos^2\frac{a}{2}-1+2\cos\frac{a}{2}\cos\frac{b-c}{2}+\frac{3}{2}\geq0$$ or
$$4\cos^2\frac{a}{2}+4\cos\frac{a}{2}\cos\frac{b-c}{2}+1\geq0$$ or
$$\left(2\cos\frac{a}{2}+\cos\frac{b-c}{2}\right)^2+\sin^2\frac{b-c}{2}\geq0,$$ which is obvious.
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