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. $\geq$ GM. (for positive numbers)
$\Rightarrow \frac{\alpha+\beta+\gamma}{3} \geq(\alpha \beta \gamma)^{1 / 3} \Rightarrow \alpha \beta \gamma \leq \frac{1}{27} \times\left(\frac{\pi^3}{8}\right)=\frac{\pi^3}{216}$
(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, $\cos (\alpha+\beta)=\sin \gamma$.