If a line makes angles `alpha, beta, gamma` with the coordinate axes, prove that `cos2alpha + cos 2beta + cos2gamma + 1 = 0`.

Explanation

`alpha , beta and gamma ` are the angles made by the line with the coordinate axes .
` :. ` We have ,
`cos^(2) alpha + cos^(2) beta + cos^(2) gamma = 1 ` ….(1)
Now, L.H.S. = ` cos 2 alpha + cos 2 beta + cos 2 gamma + 1 `
` = (2 cos^(2) alpha - 1) + (2 cos^(2) beta -1) + (2 cos^(2) gamma - 1) + 1`
` = 2 ( cos^(2) alpha + cos^(2) beta + cos^(2) gamma) - 3 + 1`
` = 2 * (1) - 2` [from (1)]
` = 2-2`
= 0= R.H.S.
` :. cos 2 alpha + cos 2 beta + cos 2 gamma + 1 = 0`.