Question

If a line makes anles `alpha, beta, gamma` with the coordinate axes, porve that `sin^2alpha+sin^2beta+sin^2gamma=2`

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 (1-sin^(2)alpha)+(1-sin^(2)beta)+(1-sin^(2)gamma)=1`
`rArrsin^(2)alpha+sin^(2)beta+sin^(2)gamma=2`.
`Hence, (sin^(2)alpha+sin^(2)beta+sin^(2)gamma)=2`.