Question:
Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.
Answer:
Given sin(A+B) = sinA*cosB + cosA*sinB
Differentiate with respect to x
cos(A+B) *(dA/dx + dB/dx) = sinA*(-sinB)dB/dx + cosB*cosA*dA/dx + cosA*cosB*dB/dx + (-sinA)*sinB*dA/dx
= (dA/dx)*[cosA*cosB - sinA*sinB ] + (dB/dx)*[cosA*cosB - sinA*sinB ]
= [cosA*cosB - sinA*sinB ]*(dA/dx + dB/dx)
=> cos(A+B) = cosA*cosB - sinA*sinB
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