Question

If a line makes angles `alpha, beta` and `gamma` with the coordinate axes, then prove that `cos 2alpha + cos2beta + cos 2gamma = -1`

Answer 1

Let the direction cosines of the given line be l,m,n. Then,
`l=cosalpha,m=cosbeta and n=cosgamma`.
`therefore (l^(2)+m^(2)+n^(2))=1rArrcos^(2)alpha+cos^(2)beta+cos^(2)gamma=1`
`rArr 2cos^(2)alpha+2cos^(2)beta+cos^(2)gamma=2`
`rArr(1+cos2alpha)+(1+cos2beta)+(1+cos2gamma)=2`
`rArr(cos2alpha+cos2beta+cos2gamma)=-1`
Hence, `(cos2alpha+cos2beta+cos2gamma)=-1`.