To understand the cosine addition formula, start with a unit radius making an angle A with the positive x-axis.
The radius has a horizontal component of length:
cos A
Now rotate the radius anticlockwise through an additional angle B. The final radius therefore makes an angle A + B with the positive x-axis.
The question is:
What does the original horizontal component, cos A, contribute to the final horizontal component?
Rotating the original horizontal component
Take the original horizontal component, whose length is cos A, and rotate it through angle B.
Its length remains cos A. After rotation, it becomes the sloping side of a right triangle.
The angle between this sloping segment and the horizontal direction is B.
Therefore, its horizontal projection is the side adjacent to B.
So:
cos B = horizontal projection / cos A
Hence:
horizontal projection = cos A × cos B
This gives the first horizontal contribution:
cos A cos B
And because this contribution points to the right, it is positive:
+cos A cos B
Connecting it to the final angle
The complete radius has now moved from angle A to angle A + B.
Therefore, its final horizontal component is:
cos(A + B)
The cos A cos B term is only the contribution coming from the original horizontal component.
There is still another horizontal contribution: the original vertical component sin A produces a leftward contribution when rotated anticlockwise. That contribution will eventually give the negative term −sin A sin B.
Thus the two contributions combine to produce:
cos(A + B) = cos A cos B − sin A sin B
Key idea
cos A is the original horizontal component. Rotating it through B leaves only a cos B fraction of it in the horizontal direction. Therefore its contribution is cos A × cos B.
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