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

If A+B+C=π,...

Question

If $A + B + C = π$ , then prove the following
$sin 2 A + sin 2 B + sin 2 C = 4 sin A . sin B . sin C$

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Solution

Given
$A + B + C = π$
Taking LHS
$sin 2 A + sin 2 B + sin 2 C$
$= 2 sin A cos A + [2 sin (B + C) cos (B - C)]$ $[∵ sin C + sin D = 2 sin (\frac{C + D}{2}) cos (\frac{C - D}{2})]$
$= 2 sin A cos A + 2 sin (B + C) cos (B - C)$
$= 2 sin A cos A + 2 sin A cos (B - C)$ $[∵ A + B + C = π]$
$= 2 sin A [- cos (B + C) + cos (B - C)]$ $[∵ A + B + C = π; cos A = cos [π - (B + C)] = - cos (B + C)]$
$= 2 sin A (2 sin B sin C)$ $[∵ cos (A - B) - cos (A + B) = 2 sin A sin B]$
$= 4 sin A sin B sin C$
$= R H S$

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, prove the following

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

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