Inequality (2)

$\textbf{In a $\mathbf{\triangle ABC},$ Prove that }$\\\\ \mathbf{\frac{1+Cos\;B\cdot Cos\;C}{1+cos \;A}+\frac{1+Cos\;C\cdot Cos\;A}{1+cos \;B}+\frac{1+Cos\;A\cdot Cos\;B}{1+cos \;C}\geq \frac{5}{2}}$

4 Answers

Devil ·

After days of waiting for an elementary solution to turn up, I finally decided to try something out-of-syllabus sort.

The point is that certain trigonometric expressions in cosA, cosB, cosC can be optimised if
the given expression can be written down as H(s,p,q)

where s=\sum {\cos A}

p=\sum {\cos A \cos B}

q=\sum {\cos A \cos B \cos C }

Needless to mention A,B,C are angles of a triangle.

Now given this I'll show how exactly the optimisation is done.

First, we will find the interior critical points of H by solving the following system of equations:

\frac{\partial H}{\partial s}+\frac{\partial H}{\partial p}s=\frac{\partial H}{\partial q}+\frac{\partial H}{\partial p}=0


\Delta =-27q^2+18spq+p^2s^2-4s^3q-4p^3>0..............(I)

Usually no such critical point exists.

Next we optimise H on degenerate triangles, i.e s=1, p=q, q\in [-1,0]


Finally optimise H over isosceles triangles - .........(III)



q=t^2(1-2t^2) : t \in [0,1]

Now applying AM-GM on the given exp. we have to analyse

\prod{\left(\frac{1+\cos A \cos B}{1+ \cos C} \right)}

Observe that this is infact


Applying the first algorithm to find the interior critical points, we have


Which is not possible since ΣcosA=2-(Σ1cosA)<-1 is a contradiction.

Region over Degenerate triangles does not yield any critical points.

And the region over isosceles triangles gives something like P(t)Q(t)-S(t)X(t)

where P\left(\frac{1}{2} \right)=X\left(\frac{1}{2} \right)=0

Clearly t=12 is a critical point where the expression gets optimised. We're yet to decide whether it's a maximum or minimum. For that we just test the given expression with A=90deg. B=45deg. C=45deg. to arrive at the conclusion that the given exp. is infact minimised at t=12.

Now obtain the values of s,p,q from (III)

Putting the respective values of s,p,q in the exp, we arrive that H(s,p,q)≥12.

An elementary soln. is very welcome. [1]

Devil ·

A small typo in the last but 1 line H(s,p,q)≥125216.

So the given expression≥3..56=52

Alternately we can straight away say that t=12 yields A=B=C=Ï€3.

And the inequality follows.

Shubhodip ·

cos(i) = 1-tan2(i/2)1+ tan2(i/2) works [6]

Devil ·

OH, maybe [3]....

Honestly I enjoyed proving this with the above stated results :D

Infact I learned that process while trying this....That's another big gain...

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