Suppose α, β, γ ∈ R are such that sin α, sin β, sin γ ≠ 0 and Δ=| sin 2 α sin α cos α cos 2 α sin 2 β sin β cos β cos 2 β sin 2 γ sin γ cos γ cos 2 γ| then Δ cannot exceed

Question

Suppose α, β, &gamma; ∈ R are such that sin α, sin β, sin &gamma; ≠ 0 and Δ=| sin 2 α sin α cos α cos 2 α sin 2 β sin β cos β cos 2 β sin 2 &gamma; sin &gamma; cos &gamma; cos 2 &gamma;| then Δ cannot exceed (A) 1 (B) 0 (C) -(1/2) (D) None of these

Tardigrade Admin · Accepted Answer

We can write Δ as, Δ= sin 2 α sin 2 β sin 2 &gamma; | 1 cot α cot 2 α 1 cot β cot 2 β 1 cot &gamma; cot 2 &gamma; | =sin 2 α sin 2 β sin 2 &gamma;( cot β- cot α) (cot &gamma;-cot α)(cot &gamma;-cot β) =sin (α-β) sin (α-&gamma;) sin (β-&gamma;) It is clear from here that Δ cannot exceed 1 . [∵ sin θ ⊁ 1, for any θ ∈ R ]

Q. Suppose $α, β, γ \in R$ are such that $sin α, sin β, sin γ \neq = 0$ and
$Δ = ∣ ∣ sin^{2} α sin^{2} β sin^{2} γ sin α cos α sin β cos β sin γ cos γ cos^{2} α cos^{2} β cos^{2} γ ∣ ∣$ then $Δ$ cannot exceed

1

0

$- \frac{1}{2}$

None of these

Q. Suppose α,β,γ∈R are such that sinα,sinβ,sinγ=0 and Δ=∣∣​sin2αsin2βsin2γ​sinαcosαsinβcosβsinγcosγ​cos2αcos2βcos2γ​∣∣​ then Δ cannot exceed

1

0

−21​

None of these

We can write Δ as, Δ=sin2αsin2βsin2γ∣∣​111​cotαcotβcotγ​cot2αcot2βcot2γ​∣∣​ =sin2αsin2βsin2γ(cotβ−cotα) (cotγ−cotα)(cotγ−cotβ) =sin(α−β)sin(α−γ)sin(β−γ) It is clear from here that Δ cannot exceed 1 . [∵sinθ⊁1, for any θ∈R]

Q. Suppose $α, β, γ \in R$ are such that $sin α, sin β, sin γ \neq = 0$ and
$Δ = ∣ ∣ sin^{2} α sin^{2} β sin^{2} γ sin α cos α sin β cos β sin γ cos γ cos^{2} α cos^{2} β cos^{2} γ ∣ ∣$ then $Δ$ cannot exceed

$- \frac{1}{2}$