The result essentially follows directly from the triple-angle identities,
sin(3A) = 3 sin(A) cos²(A) - sin³(A)
cos(3A) = cos³(A) - 3 sin²(A) cos(A)
Then
sin(3A)/sin(A) = 3 cos²(A) - sin²(A)
cos(3A)/cos(A) = cos²(A) - 3 sin²(A)
and
sin(3A)/sin(A) - cos(3A)/cos(A)
= 3(cos²(A) + sin²(A)) - (sin²(A) + cos²(A))
= 3 - 1
= 2
