Question

If sum of 3 positive numbers $\alpha ,\beta ,\gamma$ is equal to $\frac{\pi }{2}$ and $\cot \alpha ,\cot \beta ,\cot \gamma$ form an arithmetic progression then which of the following is(are) correct?

• A. Value of $\cot \alpha \cdot \cot \gamma$ is equal to 3
• B. Maximum value of the product $\alpha \beta \gamma$ is $\frac{{\pi }^3}{216}$
• C. $\cot \gamma -\tan \alpha -\tan \beta =\tan \alpha \tan \beta \cot \gamma$
• D. $\cos (\alpha +\beta )=\sin \gamma$

Answer 1

Answer: (A), (B), (C), (D)

Given $0<\alpha ,\beta ,\gamma <\frac{\pi }{2}$...(1)
Also, $2\cot \beta =\cot \alpha +\cot \gamma$....(2)
(A)$\alpha +\gamma =\left(90^{\circ }-\beta \right)$
$\Rightarrow \cot (\alpha +\gamma )=\tan \beta$
$\Rightarrow \frac{\cot \alpha \cot \gamma -1}{\cot \gamma +\cot \alpha }=\tan \beta$
$\Rightarrow \cot \alpha \cot \gamma =3\text{ [Using (2)] }$
(B) Using A.M. $\ge$ GM. (for positive numbers)
$\Rightarrow \frac{\alpha +\beta +\gamma }{3}\ge (\alpha \beta \gamma {)}^{1/3}\Rightarrow \alpha \beta \gamma \le \frac{1}{27}\times \left(\frac{{\pi }^3}{8}\right)=\frac{{\pi }^3}{216}$
(C) As, $\gamma =\left[90^{\circ }-(\alpha +\beta )\right]$
$\Rightarrow \cot \gamma =\tan (\alpha +\beta )\Rightarrow \cot \gamma =\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }$
$\Rightarrow \cot \gamma -\tan \alpha -\tan \beta =\tan \alpha \cdot \tan \beta \cdot \cot \gamma$
(D) Obviously, $\cos (\alpha +\beta )=\sin \gamma$.