If a line in 3 dimensions makes angles α , β and γ with the...

Using the trigonometric identity $\cos(A+B)\cos(A-B) = \cos^2 A - \sin^2 B$, we have:

$\cos(\alpha+\beta)\cos(\alpha-\beta) = \cos^2\alpha - \sin^2\beta$

Since $\sin^2\beta = 1 - \cos^2\beta$, the expression becomes:

$\cos^2\alpha - (1 - \cos^2\beta) = \cos^2\alpha + \cos^2\beta - 1$

For a line in three-dimensional space making angles $\alpha$, $\beta$, and $\gamma$ with the coordinate axes, the sum of the squares of its direction cosines is $1$:

$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$

$\cos^2\alpha + \cos^2\beta - 1 = -\cos^2\gamma$

Substituting this into the earlier expression, we get $-\cos^2\gamma$.

Answer: $-\cos^2\gamma$