If 0 lt α, β, γ lt (π)/2 such that α+β+γ=(π)/2 and cot α, cot β, cot γ are in AP, then the value of cot α cot γ is

Correct answer: (B) 3. Given, α+β+γ=(π)/2 ⇒ cot α+cot β+cot γ=cot α cot β cot γ ∴ cot α, cot β, cot γ are in AP. ⇒ 3 cot β=cot α cot β cot γ ⇒ cot α cot γ=3

Q90 / 170 · Manipal (MET) 2016 · Mathematics

If $0 \lt \alpha, \beta, \gamma \lt \frac{\pi}{2}$ 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
$1$
Correct answer
B
$3$
C
$\cot ^2 \beta$
D
None of these

Solution

$\begin{aligned} & \text { Given, } \alpha+\beta+\gamma=\frac{\pi}{2} \\ & \Rightarrow \cot \alpha+\cot \beta+\cot \gamma=\cot \alpha \cot \beta \cot \gamma\end{aligned}$ $\therefore \cot \alpha, \cot \beta, \cot \gamma$ are in AP. $\begin{array}{ll}\Rightarrow & 3 \cot \beta=\cot \alpha \cot \beta \cot \gamma \\ \Rightarrow & \cot \alpha \cot \gamma=3\end{array}$

Hence, the correct answer is option: 'B'.

If $0 \lt \alpha, \beta, \gamma \lt \frac{\pi}{2}$ 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

Solution

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