Cos x + y = cos x cos y - sin x sin y

In this diagram, angle CAB is denoted x, and angle DAC is denoted y.
cos x+y is AP/AD.  I will show this is the same as
cos x cos y - sin x sin y.

First I'll show that AB/AD is cos x cos y.
Then I'll show that PB/AD is sin x sin y.
Finally, since AP is AB-PB, it is clear that AP/AD is the difference of these two.

Demonstration that AB/AD is cos x cos y.

From the diagram, above, you can see:

AC = AD cos y
AB = AC cos x
AB = AD cos y cos x

Demonstration that PB/AD is sin x sin y.

I added line segment RC, perpendicular to DP.  I also extended segment BC to show the exterior angle, congruent to CAB.  Its measure is x.  Since BC is parallel to DP, angle RDC also has measure x.

CD = AD sin y
PB = RC = CD sin x
PB = AD sin y sin x

AB - PB = AD cos x cos y - AD sin x sin y
AP = AD cos x cos y - AD sin x sin y
AP = AD cos x cos y - AD sin x sin y
AP/AD =  cos x cos y - sin x sin y
cos x + y  =  cos x cos y - sin x sin y

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The webmaster and author of the Math Help site is Graeme McRae.
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