If $α + β - γ = π$ $alpha +beta -gamma =pi$ Prove that $({sin}^{2} α + {sin}^{2} β - {sin}^{2} γ) =$ $(sin^2alpha + sin^2beta - sin^2gamma)=$ 2sin alpha ×2sin beta ×2 cos gamma#?

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If $α + β - γ = π$ $alpha +beta -gamma =pi$ Prove that $({sin}^{2} α + {sin}^{2} β - {sin}^{2} γ) =$ $(sin^2alpha + sin^2beta - sin^2gamma)=$ 2sin alpha ×2sin beta ×2 cos gamma#?

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Answer 1 · 2018-07-09T03:09:41

$L H S = {sin}^{2} α + {sin}^{2} β - {sin}^{2} γ$ $LHS=sin^2alpha+sin^2beta-sin^2gamma$

$= \frac{1}{2} (1 - cos 2 α) + \frac{1}{2} (1 - cos 2 β) - \frac{1}{2} (1 - cos 2 γ)$ $=1/2(1-cos2alpha)+1/2(1-cos2beta)-1/2(1-cos2gamma)$

$= \frac{1}{2} (1 + cos 2 γ - (cos 2 α + cos 2 β))$ $=1/2(1+cos2gamma-(cos2alpha+cos2beta))$

$= \frac{1}{2} (2 {cos}^{2} γ - (2 cos (α + β) cos (α - β))$ $=1/2(2cos^2gamma-(2cos(alpha+beta)cos(alpha-beta))$

$= {cos}^{2} γ - cos (π + γ) cos (α - β)$ $=cos^2gamma-cos(pi+gamma)cos(alpha-beta)$
$= cos γ \cdot cos γ + cos γ cos (α - β)$ $=cosgamma*cosgamma+cosgammacos(alpha-beta)$

$= cos γ \cdot cos γ + cos γ cos (α - β)$ $=cosgamma*cosgamma+cosgammacos(alpha-beta)$

$= cos γ (cos (α - β) + cos (α + β - π))$ $=cosgamma(cos(alpha-beta)+cos(alpha+beta-pi))$

$= cos γ (cos (α - β) - cos (α + β))$ $=cosgamma(cos(alpha-beta)-cos(alpha+beta))$

$= cos γ \cdot 2 sin α sin β$ $=cosgamma*2sinalphasinbeta$

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If $α + β - γ = π$ $alpha +beta -gamma =pi$ Prove that $({sin}^{2} α + {sin}^{2} β - {sin}^{2} γ) =$ $(sin^2alpha + sin^2beta - sin^2gamma)=$ 2sin alpha ×2sin beta ×2 cos gamma#?

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If $α + β - γ = π$ $alpha +beta -gamma =pi$ Prove that $({sin}^{2} α + {sin}^{2} β - {sin}^{2} γ) =$ $(sin^2alpha + sin^2beta - sin^2gamma)=$ 2sin alpha ×2sin beta ×2 cos gamma#?