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Question
If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0.
Solution 1
Consider cos 2α + cos2β + cos2γ+ 1
`=(2cos^2alpha-1)+(2cos^2beta-1)+(2cos^2gamma-1)`
`=2(cos^2alpha+cos^2beta+cos^2gamma)-3`
`=2(1)-3` [∵`cos^2alpha+cos^2beta+cos^2gamma=1` ]
`=-1`
`therefore cos 2α + cos2β + cos2γ=-1`
`therefore cos 2α + cos2β + cos2γ+1=0`
Solution 2
L.H.S: cos 2α + cos2β + cos2γ + 1
`= 2cos^2alpha - 1+2cos^2beta - 1+ 2cos^2gamma-1+1`
`= 2(cos^2alpha + cos^2beta + cos^2gamma)-2`
= 2 x 1 - 2
= 2- 2
= 0
= R.H.S
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