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In any DeltaABC, prove that : (a^2 sin (...

In any `DeltaABC`, prove that : `(a^2 sin (B-C))/(sin B + sin C) + (b^2 sin (C-A))/(sin C + sin A) + (c^2 sin (A-B))/(sin A + sin B) = 0`

Text Solution

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`(a^(2) sin (B - C))/(sin B + sin C) = (4R^(2) sin ^(2) A sin (B - C))/(sin B + sin C) ( :' a = 2R sin A)`
`= (4R^(2) sin A sin (B + C) sin (B - C))/(sin B + sin C)`
`= (4R^(2) sin A (sin^(2) B - sin^(2) C))/(sin B + sin C)`
`= 4R^(2) sin A (sin B - sin C)`
(i) similarly, `(b^(2) sin (C - A))/(sin C + sin A) = 4R^(2) sin B (sin C - sin A)`
and `(c^(2) sin (A - B))/(sin A + sin B) = 4R^(2) sin C (sin A - sin B)`
Adding we get
`(a^(2) sin (B - C))/(sin B + sin C) + (b^(2) sin (C - A))/(sin C + sin A) + (c^(2) sin (A - B))/(sin A + sin B) = 0`
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Knowledge Check

  • ( cos A )/( sin B sin C ) + ( cos B )/( sin C sin A ) + ( cos C )/( sin A sin B ) =

    A
    1
    B
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    C
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    D
    none

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