tan(A + B) and tan(A - B)

Divide sin(A + B) by cos(A + B), then divide top and bottom by cos A cos B, and the tangent formulas come out in tangents alone:

tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B))

tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B))

They need tan A and tan B to exist (neither angle is 90° plus a whole number of half turns) and the bottom not to be 0. On top the sign follows the angle; below it is the opposite. For 75° as 45° + 30°, with tan 45° = 1 and tan(30°) = √(3)/3:

tan(75°) = (1 + √(3)/3)/(1 - √(3)/3) = (3 + √(3))/(3 - √(3))

A root below is cleared by multiplying top and bottom by its conjugate, here 3 + √3: the bottom becomes 9 - 3 = 6, the top (3 + √(3))² = 12 + 6√(3), and tan(75°) = 2 + √(3).