**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

### Related pages in this website

Trig Identities

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