The cosine addition formula is:

cos(A + B) = cos A cos B − sin A sin B

The two products in this formula come from the two original components of a radius.

The most important question is:

Why is the second term subtracted?

The answer comes from direction.

The second horizontal contribution points toward the negative x-direction. Therefore, although its length is positive, its contribution to the horizontal coordinate is negative.

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1. Start with a Unit Radius at Angle A

Consider a unit circle with centre O and a radius making an angle A with the positive x-axis.

Because the radius has length 1, it has two components:

Horizontal component = cos A

Vertical component = sin A

So the original radius can be viewed as one vector made from two perpendicular components:

             P
            /|
           / |
          /  | sin A
         /   |
        /    |
       O─────H
          cos A

The radius itself has not been replaced by two different radii.

We are simply describing its horizontal and vertical parts.

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2. Rotate the Radius Through Angle B

Now rotate the entire radius anticlockwise through angle B.

Its new angle becomes:

A + B

Our goal is to find the new horizontal component.

That new horizontal component will eventually be:

cos(A + B)

To find it, we examine what happens to the two original components separately.

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3. What Happens to the Original Horizontal Component?

The original horizontal component has length:

cos A

Imagine taking this length and rotating it anticlockwise through angle B.

Its length remains unchanged, but its direction changes.

After rotation, it has both a horizontal and a vertical component.

For its horizontal component, angle B is measured from the horizontal.

Therefore, the horizontal part is the adjacent side.

So:

cos B = horizontal contribution / cos A

Therefore:

horizontal contribution = cos A × cos B

This contribution points to the right.

Therefore, its signed horizontal contribution is:

+cos A cos B

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4. Now Consider the Original Vertical Component

The original vertical component has length:

sin A

Initially, it points straight upward.

Now rotate it anticlockwise through B.

Because it starts by pointing upward, anticlockwise rotation makes it point toward the upper-left.

          N
         ↖
        /
       /
      /
     O

The rotated component therefore has:

• an upward component
• a leftward horizontal component

The leftward component is the one we need to understand.

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5. Find the Length of the Leftward Component

Use the original vertical component as the radius of an auxiliary circle.

Its radius is:

sin A

After rotating through B, suppose the horizontal component has length NS.

The resulting right triangle can be represented as:

          N
          ●
         /|
        / |
       /  |
      / B |
     /    |
    C─────S

Here:

CN = sin A

and NS is the horizontal component.

Since NS is opposite angle B:

sin B = NS / CN

Substituting:

sin B = NS / sin A

Therefore:

NS = sin A × sin B

So the length of the horizontal component is:

sin A sin B

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6. But NS Points to the Left

This is the crucial step.

The length NS is positive.

However, its direction is toward the left.

On the standard coordinate system:

←────────────────O────────────────→
      negative x          positive x

Therefore:

rightward = positive

leftward = negative

So NS has a positive length but a negative x-direction.

Therefore, its signed horizontal contribution is:

−sin A sin B

This is where the minus sign in the cosine addition formula comes from.

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7. Length and Direction Are Different

This distinction is extremely important.

We should not say:

“The length of NS is negative.”

Instead:

“The length of NS is positive, but its x-direction is negative.”

For example, if a person walks 3 metres to the left:

Distance travelled = 3 metres

but:

Displacement in the x-direction = −3 metres

The same principle applies here.

Thus:

Length of NS = sin A sin B

but:

Horizontal contribution = −sin A sin B

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8. Now Combine the Two Horizontal Contributions

We have found two horizontal contributions.

From the original horizontal component

It points right:

+cos A cos B

From the original vertical component

It points left:

−sin A sin B

Therefore, the final horizontal component is:

cos A cos B − sin A sin B

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9. Why Does This Equal cos(A + B)?

After the anticlockwise rotation through B, the final radius makes an angle:

A + B

with the positive x-axis.

Because the radius is a unit radius, its horizontal component is:

cos(A + B)

But we have just calculated that same horizontal component in terms of the original components:

cos A cos B − sin A sin B

Therefore:

cos(A + B) = cos A cos B − sin A sin B

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10. The Complete Idea

The entire derivation can be summarized as follows.

Original horizontal component

cos A

After rotation through B:

+cos A cos B

because its horizontal contribution points right.

Original vertical component

sin A

After rotation through B:

−sin A sin B

because its horizontal contribution points left.

Therefore:

Final horizontal component = cos A cos B − sin A sin B

And because the final angle is A + B:

cos(A + B) = cos A cos B − sin A sin B

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The Key Insight

The minus sign is not an algebraic trick.

It comes directly from the geometry.

The original horizontal component produces a rightward contribution, while the original vertical component produces a leftward contribution. Since right is positive and left is negative on the x-axis, the two horizontal contributions are subtracted.

So the cosine addition formula can be understood as a statement about components and directions:

cos(A + B) = cos A cos B − sin A sin B

The two terms come from the two original components, and the minus sign records the fact that their horizontal contributions point in opposite directions.