Addition formulas

The addition formulas express cos(α ± β) and sin(α ± β) through the sines and cosines of α and β. cos(α + β) = cos α cos β − sin α sin β; cos(α − β) = cos α cos β + sin α sin β; sin(α + β) = sin α cos β + cos α sin β; sin(α − β) = sin α cos β − cos α sin β. The cosine formula is derived from the distance formula between two points of the unit circle; the others follow by replacing β with −β and using cos(π/2 − α) = sin α. For tangent, tan(α + β) = (tan α + tan β) : (1 − tan α tan β). These formulas help find values for angles not in the table (for example 75° = 45° + 30°, 15° = 45° − 30°) and simplify expressions. Important: cos(α + β) ≠ cos α + cos β.