If a line in 3 dimensions makes angles , and with the positive directions of the coordinate axes, then what is ( + ) ( - ) equal to?

Step-by-step solution

Using the trigonometric identity (A+B) (A-B) = ^2 A - ^2 B , we have: ( + ) ( - ) = ^2 - ^2 Since ^2 = 1 - ^2 , the expression becomes: ^2 - (1 - ^2 ) = ^2 + ^2 - 1 For a line in three-dimensional space making angles , , and with the coordinate axes, the sum of the squares of its direction cosines is 1 : ^2 + ^2 + ^2 = 1 ^2 + ^2 - 1 = - ^2 Substituting this into the earlier expression, we get - ^2 . Answer: - ^2