The Prosthaphaeresis formulas, also known as Simpson's formulas, are trigonometry formulas that convert a product of functions into a sum or difference. They are given
by

(1)

(2)

(3)

(4)

This form of trigonometric functions can be obtained in the Wolfram
Language using the command TrigFactor[expr].

These can be derived using the above figure (Kung 1996). From the figure, define

(5)

(6)

Then we have the identity

(7)

(8)

(9)

(10)

Trigonometric product formulas for the difference of the cosines and sines of two angles can be derived using the similar figure illustrated above (Kung 1996). With
and
as previously defined, the above figure gives