In the derivation of the cosine addition formula, we start with a unit radius at angle A.

Its horizontal component is:

cos A

When the entire radius is rotated anticlockwise through an additional angle B, this horizontal component also changes its direction.

The question is:

What horizontal contribution does the original cos A component make after the rotation?

The answer is:

cos A cos B

Let’s see why.

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1. Start with the original horizontal component

At angle A, the unit radius has horizontal component:

cos A

So imagine a horizontal segment of length cos A:

O────────────H
     cos A

This segment represents the original horizontal component.

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2. Rotate this component through B

Now imagine that this segment is attached at O and rotated anticlockwise through angle B.

The important thing is that rotation changes the direction, but not the length.

Therefore, after rotation, the sloping segment still has length:

cos A

We can represent it as:

             P
            /
           /   length = cos A
          /
         / B
        O────────H

The angle between the original horizontal direction and the new sloping direction is B.

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3. Find the horizontal projection

Drop a vertical line from P to the horizontal axis.

We now have a right triangle:

             P
            /|
           / |
          /  |
         / B |
        /    |
       O─────H

The horizontal side OH is the part of the rotated segment that remains in the horizontal direction.

This is the quantity we want.

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4. Identify the sides relative to B

Look at angle B.

For this right triangle:

• OP is the hypotenuse.
• OH is the adjacent side.
• PH is the opposite side.

We already know:

OP = cos A

because the segment was originally the horizontal component of length cos A.

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5. Use the definition of cosine

From the basic definition:

cos B = adjacent / hypotenuse

For our triangle:

cos B = OH / OP

Since:

OP = cos A

we get:

cos B = OH / cos A

Multiply both sides by cos A:

OH = cos A × cos B

Therefore:

The horizontal contribution of the original cos A component after rotation through B is cos A cos B.

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6. Why are we multiplying?

This is perhaps the most important part to understand.

The original segment has length:

cos A

After it is tilted through B, only a certain fraction of that length remains horizontal.

That fraction is:

cos B

So:

horizontal part

= original length × horizontal fraction

Therefore:

horizontal part

= cos A × cos B

= cos A cos B

In other words:

cos A tells us how long the original horizontal component is, while cos B tells us what fraction of that length appears horizontally after the rotation.

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7. A numerical example

Suppose, just for illustration:

cos A = 10

and:

cos B = 0.8

We therefore have a segment of length 10 tilted through angle B.

Since 0.8 of its length remains in the horizontal direction:

horizontal component = 10 × 0.8

= 8

So:

cos A cos B = 10 × 0.8 = 8

The multiplication is therefore simply scaling the original component by the appropriate fraction.

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8. A useful way to visualize it

Think of the process in two stages:

Original horizontal component

        cos A
          │
          │
          ▼
    Rotate through B
          │
          │
          ▼
Only the horizontal fraction cos B remains
          │
          ▼
     cos A × cos B

So we are not saying that cos B replaces cos A.

Instead:

cos A is the original length.

cos B tells us how much of that length appears horizontally after the rotation.

Therefore:

cos A × cos B

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9. How this fits into the cosine addition formula

We have now found one contribution to the final horizontal component:

cos A cos B

There is another contribution.

The original vertical component has length:

sin A

When it is rotated anticlockwise through B, it produces a leftward horizontal component of magnitude:

sin A sin B

Because it points left, its contribution is negative.

Therefore the final horizontal component is:

cos A cos B − sin A sin B

But the final radius is at angle:

A + B

So its horizontal component is:

cos(A+B)

Therefore:

cos(A+B) = cos A cos B − sin A sin B

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Key takeaway

The term cos A cos B has a very simple geometric meaning:

Start with the original horizontal component cos A. Rotate it through B. The fraction of that component that remains horizontal is cos B. Therefore its new horizontal contribution is cos A × cos B.

This is the component technique that allows the cosine addition formula to be understood geometrically rather than memorized.