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For the triangle $\triangle ABC$, prove the inquality, $$\frac1{4+\cos A\cos(B-C)}+\frac1{4+\cos B\cos(C-A)}+\frac1{4+\cos C\cos(A-B)}\ge \frac23$$ where $A$, $B$ and $C$ are the vertex angles.

I had trouble tackling it. Have treated it as a minimization problem, subject to the constraint $A+B+C=\pi$. In principle, it could be solved with the Lagrange multiplier. However, the required algebraic work seems to be prohibitive. Wonder whether there is a more elegant path to show it.

Blue
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Quanto
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2 Answers2

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First, we use the trig identity $ \cos A \cos B = \frac{1}{2} (\cos(A+B) + \cos(A-B)) $:

$$\frac1{4 + \cos A \cos(B-C)}+\frac1{4 +\cos B \cos(C-A)} + \frac1{4 + \cos C \cos(A-B)} \ge \frac23$$ $$ \Leftrightarrow \frac1{4 + \frac{1}{2} (\cos(A+B-C) + \cos(A-B+C))} + \frac1{4 + \frac{1}{2} (\cos(B+C-A) + \cos(B-C+A))} + \frac1{4 + \frac{1}{2} (\cos(C+A-B) + \cos(C-A+B))} \ge \frac23 $$

Since $A + B + C = \pi$, we get:

$$ \frac1{4 + \frac{1}{2} (\cos(\pi-2C) + \cos(\pi-2B))} + \frac1{4 + \frac{1}{2} (\cos(\pi-2A) + \cos(\pi-2C))} + \frac1{4 + \frac{1}{2} (\cos(\pi-2B) + \cos(\pi-2A))} \ge \frac23 $$

Using the trig identity $ \cos(\pi-x) = - \cos x$:

$$ \frac1{4 - \frac{1}{2} (\cos(2C) + \cos(2B))} + \frac1{4 - \frac{1}{2} (\cos(2A) + \cos(2C))} + \frac1{4 - \frac{1}{2} (\cos(2B) + \cos(2A))} \ge \frac23 $$

Applying the inequality $ \frac1a + \frac1b + \frac1c \ge \frac9{a+b+c}$:

$$ \frac1{4 - \frac{1}{2} (\cos(2C) + \cos(2B))} + \frac1{4 - \frac{1}{2} (\cos(2A) + \cos(2C))} + \frac1{4 - \frac{1}{2} (\cos(2B) + \cos(2A))} $$

$$ \ge \frac9{12 - (\cos(2A) + \cos(2B) + \cos(2C))} $$

Finally, use the inequality $ \cos(2A) + \cos(2B) + \cos(2C) \ge - \frac{3}{2}$ (which you can find the proof in here: Prove that $\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}$ for angles of a triangle):

$$ \frac{9}{12 - (\cos(2A) + \cos(2B) + \cos(2C))} \geq \frac{9}{12 - (- \frac{3}{2})} = \frac23 $$

The inequality holds when

$$A = B = C = \frac{\pi}{3} $$

or ABC is an equilateral triangle

The 2nd
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Using $\cos(B+C)=\cos(\pi-A)=?$

and Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$

$$2(4+\cos A\cos(B-C))$$ $$=8-\cos(B+C)\cos(B-C)$$ $$=8-(\cos^2B-\sin^2C)$$ $$=9-(\cos^2B+\cos^2C)$$

Now using AM -HM inequity,

$$\dfrac{\sum_{\text{cyc}}\dfrac2{9-(\cos^2 B+\cos^2C)}}3$$

$$\ge\dfrac3{\dfrac{27-2(\cos^2A+\cos^2B+\cos^2C)}2}$$

The equality will occur if $9-(\cos^2 B+\cos^2C)=9-(\cos^2C+\cos^2A)=9-(\cos^2A+\cos^2B)$

$\iff\cos^2A=\cos^2B=\cos^2C$

$\iff\sin^2A=\sin^2B=\sin^2C$

which will occur if $A=B=C$

using Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $

lab bhattacharjee
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