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Sin(A+B)Sin(A-B)

sin 2α -β+sin...

Question

$sin 2 (α - β) + sin 2 (β - γ) + sin 2 (γ - α)$ $= - 4 sin (α - β) sin (β - γ) sin (γ - α)$ .

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Solution

Let $α - β = x, β - γ = y, γ - α = z$ .
$x + y + z = 0$ or $x + y = - z$
$∴ sin (x + y) = - sin z, cos (c + y) = cos z$ . $(1)$
Hence we have to prove that
$sin 2 x + sin 2 y + sin 2 z = - 4 sin x sin y sin z$ .
L.H.S. $= 2 sin (x + y) cos (x - y) + 2 sin z cos z$ by $(1)$
$= - 2 sin z cos (x - y) + 2 sin z cos z$
$= - 2 sin z [cos (x - y) - cos z]$
$= - 2 sin z [cos (x - y) - cos (x + y)],$ by $(1)$
$= - 2 sin z (2 sin x sin y)$ .
$= - 4 sin x sin y sin z$ .
$= - 4 sin (α - β) sin (β - γ) sin (γ - α)$ .

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Sin(A+B)Sin(A-B)

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