How do you prove that #cos(x-y) = cosxcosy + sinxsiny#?
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
Therefore, in triangle XYQ
Similarly in triangle QZP
Since WZ is parallel to XY (by construction)
angle XPW = angle PXY =
From the diagram
Part 2 : Show that if
so we can substitute to get
By the CAST quadrant diagram for trig. signs (below) we can see that
Therefore, we can write:
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