Show that no line in space can make angles `(pi)/(6)` and `(pi)/(4)` with x -axis and y -axis.

Explanation

Let, if possible a line makes angle `(pi)/(6) and (pi)/(4)` with X-axis and Y-axis respectively.
If `alpha, beta, gamma` are the angles made by the line with the coordinate axes, then
`alpha = (pi)/(6) and beta = (pi)/(4)`
`therefore cos alpha = cos ""(pi)/(6) = (sqrt(3))/(2)` and ` cos beta = cos "" (pi)/(4) = (1)/(sqrt(2))`
Now, `cos^(2)alpha + cos^(2)beta + cos^(2)gamma = 1`
`therefore((sqrt(3))/(2))^(2)+((1)/(sqrt(2)))^(2) + cos^(2)gamma =1`
`therefore (3)/(4)+(1)/(2)+cos^(2)gamma=1`
`therefore cos^(2)gamma = 1-(5)/(4) = -(1)/(4)`
This is not possible.
Hence, no line in space can make angles `(pi)/(6) and (pi)/(4)` with X-axis and Y-axis repectively.