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How can I graphically show/prove that cos(A+B) is equal to cosA cosB + sinA sinB?

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1 Answer

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Draw a unit circle. Label the origin $O$, and a point on the $+x$ axis $X$. Draw an angle $A$ that passes through point $P$ on the unit circle, and above it an angle $B$ that passes through point $Q$, so that the total angle from the $x$ axis is $A+B$. (My picture works most easily with two acute angles that sum to less than $\pi/2$.)

Now drop a perpendicular from $OQ$ onto $OP$ at point $R$, and drop a perpendicular from $OR$ to $OX$ at point $S$. Drop a perpendicular from $OQ$ all the way to $OX$ at $T$.

From there, you can work out the lengths of many sides of triangles, giving you products of trigonometric functions, so you can see how to add and subtract distances to get what you need.

As a bonus, the same diagram proves the formula for $\sin (A+B)$ as well.

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