Prosthaphaeresis Formulas

The prosthaphaeresis formulas, also known as Simpson's formulas, are trigonometry formulas that convert a sum or difference of sines or cosines into a product. 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

			(11)
			(12)
			(13)
			(14)