The formula
cos(x + y) = cos x cos y − sin x sin y
is one of the most important identities in trigonometry.
At first, it may look like a formula that simply needs to be memorized. But the formula has a clear geometric meaning. We can derive it by starting with a vector at angle x, separating it into its horizontal and vertical components, and then seeing what each component contributes when the vector is rotated through an additional angle y.
The key idea is:
We calculate the new horizontal contribution of each original component separately and then combine those contributions.
This also explains why we do not write an extra cos x in the final formula.
1. Start with a unit vector at angle x
Consider a unit vector making an angle x with the positive horizontal axis.
Because the vector has length 1, its coordinates are:
(cos x, sin x)
Therefore, it has two components:
Horizontal component = cos x
Vertical component = sin x
We can visualize this as:
P
●
/|
/ |
/ | sin x
/ |
/ |
/ |
/ x |
/ |
O───────●
cos x
So the original vector consists of:
- a horizontal component of length cos x
- a vertical component of length sin x
We will examine these two components separately.
2. Now rotate the vector by an additional angle y
Suppose the entire vector is rotated counterclockwise through an angle y.
The original angle was:
x
The additional rotation is:
y
Therefore, the final angle is:
x + y
The final unit vector therefore has a horizontal component equal to:
cos(x + y)
Our goal is to determine this final horizontal component.
Instead of calculating it all at once, we will ask:
How much does the original horizontal component contribute?
and then:
How much does the original vertical component contribute?
3. First component: the original horizontal length cos x
Let’s temporarily ignore the vertical component.
The original horizontal component has length:
cos x
Initially, it points directly to the right:
O────────────────────→
cos x
Now rotate this component counterclockwise through angle y.
It becomes a slanted vector:
B
●
/|
/ |
/ |
/ |
/ |
/ y |
/ |
O──────A────────→
The slanted vector OB has the same length as before.
Therefore:
OB = cos x
But it is no longer completely horizontal.
We want to find its new horizontal projection OA.
4. Why is the new horizontal contribution cos x cos y?
Look at the right triangle:
B
●
/|
/ |
/ |
/ |
/ |
/ y |
/ |
O──────A
The hypotenuse is:
OB = cos x
The angle is:
y
The horizontal projection OA is adjacent to angle y.
From the definition of cosine:
cos y = adjacent side / hypotenuse
Therefore:
cos y = OA / cos x
Multiply both sides by cos x:
OA = cos x cos y
Therefore, the original horizontal component contributes:
+cos x cos y
to the final horizontal direction.
5. What does cos x cos y mean intuitively?
The expression has a very simple interpretation.
The original horizontal component has length:
cos x
When it is rotated through y, only part of that length appears horizontally.
The fraction that appears horizontally is:
cos y
Therefore:
new horizontal contribution
= original length × horizontal fraction
= cos x × cos y
= cos x cos y
So cos x cos y is not an additional length that we add to the original cos x.
It is the new horizontal projection of the original cos x component.
6. Why don’t we write cos x + cos x cos y?
This is a very important point.
Initially:
horizontal component = cos x
After rotation:
horizontal contribution from that component = cos x cos y
The original cos x was the horizontal projection before rotation.
After rotation, the vector has changed direction, so its horizontal projection has changed.
For example, imagine a horizontal vector of length 10:
O────────────────→
10
Its horizontal component is 10.
Now rotate it by 60°:
B
/
/ 10
/
O────────A
5
Since:
cos 60° = 1/2
the new horizontal projection is:
10 × 1/2 = 5
We would not calculate:
10 + 5
The 10 was the old horizontal projection. The 5 is the new horizontal projection.
Likewise:
cos x
is the old projection, while:
cos x cos y
is the new projection.
Therefore, we use:
cos x cos y
rather than:
cos x + cos x cos y
7. Second component: the original vertical length sin x
Now forget about the original horizontal component.
Let’s examine the vertical component.
Its length is:
sin x
Initially, it points straight upward:
↑
│
│
│
│ sin x
│
O────────→
Because it is completely vertical, its initial horizontal contribution is:
0
Now rotate this vertical component counterclockwise through angle y.
It tilts toward the left:
↖
/
/
/
/
/
O────────→
Therefore, it now develops a horizontal component pointing to the left.
8. Find the horizontal contribution of sin x
The original vertical component has length:
sin x
After rotating it through y, its horizontal component has magnitude:
sin x sin y
Why?
Because the horizontal component is opposite angle y.
Using:
sin y = opposite side / hypotenuse
we have:
sin y = horizontal component / sin x
Therefore:
horizontal component = sin x sin y
But this component points left.
Since right is positive and left is negative, its signed contribution is:
−sin x sin y
9. Now combine the two contributions
We have analyzed the two original components independently.
Contribution from the original horizontal component
+cos x cos y
Contribution from the original vertical component
−sin x sin y
Therefore, the total final horizontal component is:
cos x cos y − sin x sin y
But the entire vector has been rotated from angle x through an additional angle y.
So its final angle is:
x + y
The horizontal component of a unit vector at angle x + y is:
cos(x + y)
Therefore:
cos(x + y) = cos x cos y − sin x sin y
10. The complete picture
The derivation can be summarized as:
Original vector
↗
/|
/ |
/ | sin x
/ |
/ |
/ |
/ x |
O───────●
cos x
Break it into two components.
Component 1: cos x
After rotation by y:
Original length: cos x
cos x
↓
rotate by y
↓
horizontal projection
cos x cos y
Contribution:
+cos x cos y
Component 2: sin x
After rotation by y:
Original length: sin x
sin x
↓
rotate by y
↓
leftward horizontal projection
sin x sin y
Contribution:
−sin x sin y
Now combine them:
+cos x cos y
−sin x sin y
Therefore:
Final horizontal component = cos x cos y − sin x sin y
And because the final angle is x + y:
Final horizontal component = cos(x + y)
Hence:
cos(x + y) = cos x cos y − sin x sin y
11. The key idea behind the formula
The formula is much easier to understand when we don’t treat its four terms as mysterious pieces.
Each part has a specific meaning.
cos x
= size of the original horizontal component
cos y
= horizontal fraction of that component after rotation
Therefore:
cos x cos y
= new horizontal contribution from the original horizontal component
Similarly:
sin x
= size of the original vertical component
sin y
= horizontal fraction produced by rotating that vertical component
Therefore:
sin x sin y
= magnitude of its horizontal contribution
The contribution points left, so:
−sin x sin y
12. Why the two contributions can be calculated separately
This is an important mathematical principle.
The original vector can be decomposed into:
horizontal component + vertical component
That is:
cos x component + sin x component
When the vector is rotated, we can mathematically track what happens to each component separately.
The final vector is obtained by adding the rotated components.
Therefore, we can calculate:
Contribution from cos x
and:
Contribution from sin x
independently.
Only after finding both do we combine them.
This is why the derivation naturally produces:
cos x cos y − sin x sin y
13. A useful way to think about the entire process
Imagine that the original vector has two “contributors” to its final horizontal length.
Contributor 1
The original horizontal component:
cos x
After rotation:
+cos x cos y
Contributor 2
The original vertical component:
sin x
After rotation:
−sin x sin y
Then:
Final horizontal length
= first contribution + second contribution
= cos x cos y − sin x sin y
But the final horizontal length is:
cos(x + y)
Therefore:
cos(x + y) = cos x cos y − sin x sin y
14. The deeper lesson
The most important thing to understand is not simply the formula itself.
It is the process behind the formula:
Start with a vector at angle x
↓
Break it into horizontal and vertical components
↓
Horizontal component = cos x
Vertical component = sin x
↓
Rotate the vector by y
↓
Find the new horizontal contribution of cos x
= cos x cos y
↓
Find the new horizontal contribution of sin x
= −sin x sin y
↓
Add the two contributions
= cos x cos y − sin x sin y
↓
Recognize the final horizontal component
= cos(x + y)
Therefore:
cos(x + y) = cos x cos y − sin x sin y
This component-by-component viewpoint turns the angle-addition formula from a formula to memorize into a formula that can be understood and reconstructed from geometry.
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