Once the addition formulas have been understood geometrically, the subtraction formulas do not require an entirely new derivation.

The key idea is simple:

Subtracting B from A is the same as adding the opposite angle, −B, to A.

So the subtraction formulas can be obtained directly from the addition formulas by replacing B with −B.

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1. Start with the addition formulas

The two addition formulas are:

These formulas describe what happens when we start at angle A and rotate anticlockwise through B.

The final angle is:

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2. What does A − B mean geometrically?

Instead of rotating anticlockwise through B, suppose we rotate clockwise through B.

A clockwise rotation of B is represented mathematically as:

Therefore:

This is the key connection.

We don’t need a completely new component derivation.

We can use the derivation we already have for addition and simply replace B by −B.

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3. Apply this to the sine formula

Start with:

Replace B with −B:

Now we use two important properties of sine and cosine.

Cosine is an even function:

Sine is an odd function:

Therefore:

So we obtain:

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4. Apply the same idea to cosine

Start with:

Replace B with −B:

Again:

and:

Therefore:

So:

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5. What changes when we change B to −B?

This becomes especially intuitive when we connect it with the component method.

For A + B, we rotate anticlockwise:

A → A + B

For A − B, we rotate clockwise:

A → A − B

The direction of the rotation has changed.

This affects the components involving sin B.

But cosine does not change sign:

while sine changes sign:

That is why the signs of the appropriate terms change.

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6. The unit-circle picture

Imagine the radius initially pointing at angle A.

Addition

Rotate anticlockwise through B:

              final
                ↗
               /
              / B
             /
            / A
           O────────→

The final angle is:

A + B

Subtraction

Rotate clockwise through B:

           A
          /
         /
        O────────→
         \
          \ B
           ↘
          final

The final angle is:

A − B

The two situations use the same original radius and the same Cartesian coordinate system.

Only the direction of the additional rotation changes.

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7. Why cosine keeps the same value for ±B

There is a useful geometric reason for:

Angles B and −B are mirror images across the x-axis.

Their points on the unit circle have the same horizontal coordinate.

For example:

              •
             / B
            /
O──────────┼──────────→
            \
             \ −B
              •

The two points are at the same horizontal distance from the y-axis.

Therefore their cosine values are equal.

So:

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8. Why sine changes sign

The vertical coordinates of those two points are opposite.

The point at +B is above the x-axis.

The point at −B is below the x-axis.

Therefore:

So:

Cosine keeps its sign when the angle is reversed, while sine changes its sign.

This is exactly what we need when converting the addition formulas into subtraction formulas.

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9. The four formulas together

We can now see all four formulas as one connected family.

Addition

Subtraction

The subtraction formulas are therefore not independent formulas that need to be memorized separately.

They follow from the addition formulas.

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The Key Insight

The geometric component approach gives us the addition formulas by examining what happens when a radius at angle A is rotated through B.

Once that is understood, subtraction is simply the same process in the opposite direction.

A − B means A + (−B).

So we can use the addition formulas with −B.

Because:

and:

the appropriate signs automatically change.

This gives:

and:

The important lesson is therefore not to memorize four unrelated formulas, but to understand the underlying idea:

Addition corresponds to rotating through B in one direction; subtraction corresponds to rotating through −B in the opposite direction.