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Prove that : cos^2 (beta-gamma) + cos^2 ...

Prove that : `cos^2 (beta-gamma) + cos^2 (gamma-alpha) + cos^2 (alpha-beta) =1+2cos (beta-gamma) cos (gamma-alpha) cos (alpha-beta)`.

Text Solution

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Let `A=beta-gamma,B=gamma-a`, and `C-pi=alpha-beta`
`rArr A+B+C=pi`
`rArr cos^(2)(beta-gamma)+cos^(2)(gamma-alpha)+cos^(2)(alpha-beta)`
`=cos^(2)A+cos^(2)B+cos^(2)C`
`=(3+(cos2A_cos 2B+cos2C))/(2)`
`=1-2cos A cos B cos C`
`=1+2 cos (beta-alpha)cos(gamma-alpha)cos(alpha-beta)`
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