Derivation of Addition, Subtraction, and Multiple Angle Formulas

Let P1 (x1 , y2) and P2 (x2 , y2) be arbitrary points on the Unit Circle as pictured on the left. Let α1 and α2 be the angles in standard position associated P1 and P2 respectively. Let P3 (x3 , y3) be the point in standard position associated with α3 = α1 - α2 .

Notice the distance between P1 and P2 is the same as the distance between P3 and (1, 0). Let’s use this observation to derive a trigonometry identity.
d(P1, P2) = d(P3, (1,0))

Write the distance formula and then click here to check your work.





Now notice: (Click on the blank line to see answers after you fill in the blank.)

x1 = cos α1,   x2 =                     ,   x3 =                     ,   y1 =                     ,   y2 =                     ,   y3 =                     ,  

Get rid of the x's and y's in the distance formula and then click here.





Replace α3 with α1 − α2 and solve for cos(α1 − α2):





Now we will use the above to derive cos(α1 + α2):





Next let’s write cos(π/2 − α) and sin(π/2 − α) in terms of sin α and cos α.





Thus