Answer :

Given, the line makes the angles α, β and γ respectively with x-axis, y-axis and z-axis.


As per the relation between direction cosines of a line, l2 + m2+n2 = 1 where l,m,n are the direction cosines of a line from x-axis, y-axis and z-axis respectively.


So, we can say that,


cos2α + cos2β + cos2γ = 1 ------ (1)


Now, we should find the value for


cos2α + cos2β + cos2γ


cos2α can be written as 2cos2α -1,


cos2α + cos2β + cos2γ = (2cos2α -1) + (2cos2β -1) + (2cos2γ -1)


= 2 (cos2α+ cos2β + cos2γ) – 3


= 2(1) – 3


[From Equation (1)]


= -1


Therefore,


cos2α + cos2β + cos2γ = -1




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