Question

If $0<\alpha, \beta, \gamma<\frac{\pi}{2}$ such that $\alpha+\beta+\gamma=\frac{\pi}{2}$ and $\cot \alpha, \cot \beta, \cot \gamma$ are in AP, then the value of $\cot \alpha \cot \gamma$ is

• A. 1
• B. 3
• C. $\cot ^{2} \beta$
• D. None of these

Answer 1

Answer: (B)

Given, $\alpha+\beta+\gamma=\frac{\pi}{2}$
$\Rightarrow \cot \alpha+\cot \beta+\cot \gamma=\cot \alpha \cot \beta \cot \gamma$
$\therefore \cot \alpha, \cot \beta, \cot \gamma$ are in AP.
$\Rightarrow 3 \cot \beta=\cot \alpha \cot \beta \cot \gamma$
$\Rightarrow \cot \alpha \cot \gamma=3$