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If a line makes angles alpha, beta, gamm...

If a line makes angles `alpha, beta, gamma` with the coordinate axes, prove that `cos2alpha + cos 2beta + cos2gamma + 1 = 0`.

Text Solution

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LHS = `cos2 alpha + cos 2beta + cos2gamma + 1`
` = 2 cos^(2) alpha = 1 + 2cos^(2) beta -1 + 2 cos^(2) gamma -1 + 1`
` = 2(cos^(2) alpha + cos^(2) beta + cos^(2)gamma) -2`
` = 2(1) -2 " "…[because cos^(2)alpha + cos^(2) beta + cos^(2)gamma = 1]`
` = 0 =` RHS.
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