Using the fact that sin(A + B) = sinAcosB + cosAsinB and the differentiation, obtain the sum formula for cosines.

To derive the sum formula for cosines (i.e., the formula for cos(A + B)), we can use the following trigonometric identity:

sin(A + B) = sinA * cosB + cosA * sinB

Now, let's differentiate both sides of this equation with respect to A:

d/dA [sin(A + B)] = d/dA [sinA * cosB + cosA * sinB]

Using the chain rule, we can differentiate sin(A + B) with respect to A as follows:

cos(A + B) * (d/dA)(A + B) = sinA * cosB + cosA * sinB

Now, simplify the left side:

cos(A + B) * 1 = sinA * cosB + cosA * sinB

cos(A + B) = sinA * cosB + cosA * sinB

This is the sum formula for cosines:

cos(A + B) = cosA * cosB - sinA * sinB

So, using the fact that sin(A + B) = sinA * cosB + cosA * sinB and differentiation, we have derived the sum formula for cosines, which is cos(A + B) = cosA * cosB - sinA * sinB.