1. Tardigrade
  2. Question
  3. Mathematics
  4. 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

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Q. Suppose $\alpha, \beta, \gamma \in R$ are such that $\sin \alpha, \sin \beta, \sin \gamma \neq 0$ and
$\Delta=\begin{vmatrix}\sin ^{2} \alpha & \sin \alpha \cos \alpha & \cos ^{2} \alpha \\ \sin ^{2} \beta & \sin \beta \cos \beta & \cos ^{2} \beta \\ \sin ^{2} \gamma & \sin \gamma \cos \gamma & \cos ^{2} \gamma\end{vmatrix}$ then $\Delta$ cannot exceed

Determinants

Solution:

We can write $\Delta$ as,
$\Delta=\sin ^{2} \alpha \sin ^{2} \beta \sin ^{2} \gamma \begin{vmatrix} 1 & \cot \alpha & \cot ^{2} \alpha \\ 1 & \cot \beta & \cot ^{2} \beta \\ 1 & \cot \gamma & \cot ^{2} \gamma \end{vmatrix}$
$=sin ^{2} \,\alpha \,sin ^{2} \beta \,sin ^{2} \gamma(\cot \beta-\cot \alpha)$
$(cot \,\gamma-cot\, \alpha)(cot \,\gamma-cot \,\beta)$
$=sin (\alpha-\beta) \sin (\alpha-\gamma) \sin (\beta-\gamma)$
It is clear from here that $\Delta$ cannot exceed 1 .
$[\because \sin \theta ⊁ 1$, for any $ \theta \in R ]$