Maharashtra State BoardHSC Arts 12th Board Exam
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If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0. - Mathematics and Statistics

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Sum

If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0.

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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

Concept: Trigonometric Functions - Trigonometric equations
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