# How do you prove that #cos(x-y) = cosxcosy + sinxsiny#?

##### 1 Answer

can be demonstrated by first showing that

and then doing the conversion using the CAST principle as indicated.

- I'm sure there are other ways to do this; but this is what I came up with. (it is pretty long).
- My apologies for using
#a# and#b# instead of#x# and#y# ; I drew the diagrams below before checking what variables had been used in the request.

**Part 1 :** Show

A triangle XQP has been constructed along the hypotenuse of triangle XYQ with angle

The line segment XP is identified as the unit length for all measurements in this system.

A rectangle is constructed with base XY by extending the line from Y through Q until a point Z is reached where PZ is parallel to the bottom (XY) (completion of the rectangle establishes point W)

Within triangle XQP is is clear that (since

and

Therefore, in triangle XYQ

Similarly in triangle QZP

Since WZ is parallel to XY (by construction)

angle XPW = angle PXY =

and

From the diagram

or

**Part 2** : Show that if

then

so we can substitute to get

By the CAST quadrant diagram for trig. signs (below) we can see that

and

Therefore, we can write:

or