If 0

Question

If 0<α, β, &gamma;<(π/2) such that α+β+&gamma;=(π/2) and cot α, cot β, cot &gamma; are in AP, then the value of cot α cot &gamma; is (A) 1 (B) 3 (C)  cot 2 β (D) None of these

Tardigrade Admin · Accepted Answer

Given, α+β+&gamma;=(π/2) ⇒ cot α+ cot β+ cot &gamma;= cot α cot β cot &gamma; ∴ cot α, cot β, cot &gamma; are in AP. ⇒ 3 cot β= cot α cot β cot &gamma; ⇒ cot α cot &gamma;=3

Q. If $0 < α, β, γ < \frac{π}{2}$ such that $α + β + γ = \frac{π}{2}$ and $cot α, cot β, cot γ$ are in AP, then the value of $cot α cot γ$ is

1

3

$cot^{2} β$

None of these

Given, $α + β + γ = \frac{π}{2}$
$\Rightarrow cot α + cot β + cot γ = cot α cot β cot γ$
$∴ cot α, cot β, cot γ$ are in AP.
$\Rightarrow 3 cot β = cot α cot β cot γ$
$\Rightarrow cot α cot γ = 3$

Q. If 0<α,β,γ<2π​ such that α+β+γ=2π​ and cotα,cotβ,cotγ are in AP, then the value of cotαcotγ is

1

3

cot2β

None of these

Given, α+β+γ=2π​ ⇒cotα+cotβ+cotγ=cotαcotβcotγ ∴cotα,cotβ,cotγ are in AP. ⇒3cotβ=cotαcotβcotγ ⇒cotαcotγ=3

Q. If $0 < α, β, γ < \frac{π}{2}$ such that $α + β + γ = \frac{π}{2}$ and $cot α, cot β, cot γ$ are in AP, then the value of $cot α cot γ$ is

$cot^{2} β$

Given, $α + β + γ = \frac{π}{2}$
$\Rightarrow cot α + cot β + cot γ = cot α cot β cot γ$
$∴ cot α, cot β, cot γ$ are in AP.
$\Rightarrow 3 cot β = cot α cot β cot γ$
$\Rightarrow cot α cot γ = 3$