An Inequality with Angles and Integers

Problem

An  Inequality with Angles and Integers

Solution 1

By the AM-GM inequality,

$\displaystyle \begin{align} k^2\cos^2 \beta +l^2\cos^2 \alpha &\ge 2\sqrt{k^2l^2\cos^2\alpha \cos^2 \beta}\\ &=2|kl|\cos \alpha\cos \beta \geq 2kl \cos \alpha \cos \beta. \end{align}$

Thus, we have a sequence of equivalent inequalities:

$\displaystyle \begin{align} &k^2\cos^2 \beta+l^2\cos^2 \alpha -2kl \cos \alpha \cos \beta \geq 0\\ &\frac{k^2\cos^2 \beta}{\sin (\alpha+\beta)\cos \alpha \cos \beta}+\frac{l^2\cos^2 \alpha}{\sin (\alpha+\beta)\cos \alpha \cos \beta}-\frac{2kl \cos \alpha \cos \beta}{\sin (\alpha+\beta)\cos \alpha \cos \beta}\geq 0\\ &\frac{k^2\cos \beta}{\sin (\alpha+\beta)\cos \alpha}+\frac{l^2 \cos \alpha}{\sin (\alpha+\beta)\cos \beta}-\frac{2kl}{\sin (\alpha+\beta)}\geq 0\\ &\frac{k^2\cos (\alpha+\beta-\alpha)}{\sin (\alpha+\beta)\cos \alpha}+\frac{l^2\cos (\alpha+\beta -\beta)}{\sin (\alpha+\beta)\cos \beta}-\frac{2kl}{\sin (\alpha+\beta)}\geq 0\\ &k^2\Bigr(\frac{\sin \alpha}{\cos \alpha}+\frac{\cos (\alpha+\beta)}{\sin (\alpha+\beta)}\Bigr)+l^2 \Bigr(\frac{\sin \beta}{\cos \beta}+\frac{\cos (\alpha+\beta)}{\sin (\alpha+\beta)}-\frac{2kl}{\sin (\alpha+\beta)}\geq 0\\ &k^2 \Bigr(\tan \alpha+\cot (\alpha+\beta)\Bigr)+l^2\Bigr(\tan \beta+\cot (\alpha+\beta)\Bigr)\geq \frac{2kl}{\sin (\alpha+\beta)}\\ &k^2 \tan \alpha+l^2 \tan \beta \geq \frac{2kl}{\sin (\alpha+\beta)}-(k^2+l^2)\cot (\alpha+\beta). \end{align}$

Solution 2

$\displaystyle \begin{align} &k^2\tan\alpha + l^2\tan\beta\geq \frac{2kl}{\sin(\alpha+\beta)}-(k^2+l^2)\cot(\alpha+\beta)\,\Leftrightarrow\\ &\left(k^2\frac{\sin\alpha}{\cos\alpha}+l^2\frac{\sin\beta}{\cos\beta}\right)\sin(\alpha+\beta)+(k^2+l^2)\cos(\alpha+\beta) \geq 2kl\,\Leftrightarrow\\ &\left(k^2\frac{\sin\alpha}{\cos\alpha}+l^2\frac{\sin\beta}{\cos\beta}\right)(\sin\alpha\cos\beta+\cos\alpha\sin\beta)\\ &~~~~~~~~~~~~~~~+(k^2+l^2)(\cos\alpha\cos\beta-\sin\alpha\sin\beta)\geq 2kl\,\Leftrightarrow\\ &k^2\left(\frac{\sin^2\alpha\cos\beta}{\cos\alpha}+\cos\alpha\cos\beta\right)+l^2\left(\frac{\sin^2\beta\cos\alpha}{\cos\beta}+\cos\alpha\cos\beta\right)\geq 2kl\,\Leftrightarrow\\ &\left(k^2\frac{\cos\beta}{\cos\alpha}\right)(\sin^2\alpha+\cos^2 \alpha)+\left(l^2\frac{\cos\alpha}{\cos\beta}\right)(\sin^2\beta+\cos^2 \beta) \geq 2kl\,\Leftrightarrow\\ &k^2\frac{\cos\beta}{\cos\alpha}+l^2\frac{\cos\alpha}{\cos\beta}\geq 2kl. \end{align}$

Note that $\cos\alpha$, $\cos\beta$ and $\sin(\alpha+\beta)$ are positive over the domain defined in the problem. Thus, the last inequality follows from AM-GM and the first inequality can be derived from the last inequality by reversing all the steps.

Acknowledgment

Dan Sitaru has kindly emailed me a LaTeX file with his solution (Solution 1) to the above problem, originally from the School Science and Mathematics Association. The problem is by ARKADY ALT, SAN JOSE, CA. Solution 2 is by Amit Itagi.

 

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