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Question
Show that no line in space can make angles `pi/6` and `pi/4` with X-axis and Y-axis.
Solution
Let, if possible, a line in space make angles `pi/6 and pi/4` with X-axis and Y-axis.
∴ α = `pi/6, beta = pi/4`
Let the line make angle γ with Z-axis
∵ cos2α + cos2β + cos2γ = 1
∴ `"cos"^2(pi/6) + "cos"^2(pi/4) + "cos"^2gamma = 1`
∴ `(sqrt3/2)^2 + (1/sqrt2)^2 + "cos"^2gamma = 1`
∴ `"cos"^2gamma = 1 - 3/4 - 1/2 = - 1/4`
This is not possible, because cos γ is real.
∴ cos2γ cannot be negative.
Hence, there is no line in space which makes angles `pi/6 and pi/4` with X-axis and Y-axis.
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