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If alpha + beta = gamma prove that cos^...

If `alpha + beta = gamma` prove that `cos^2 alpha+ cos^2 beta+cos^2gamma=1+2cos alpha cos beta cos gamma`

Text Solution

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We have to prove,
`cos^2alpha+cos^2beta+cos^2gamma = 1+2cosalphacosbetacosgamma`
We can write it as
`cos^2alpha+cos^2beta+cos^2gamma -1 = 2cosalphacosbetacosgamma`
`L.H.S. = cos^2alpha+cos^2beta+cos^2gamma -1`
`= cos^2alpha+cos^2beta-1(1-cos^2gamma)`
`= cos^2alpha+cos^2beta-sin^2gamma`
`= cos^2alpha+cos^2beta-sin^2(alpha+beta)` (As `gamma = alpha+beta`)
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