You must already know the addition formula and subtraction formulas for sine and cosine. Now suppose I do [sin(j + k) + sin(j - k)]:
If we cancel the like terms we get:
That is the product rule for [sin(j) * cos(k)]. Now suppose I do [cos(j + k) + cos(j - k)]:
If we cancel out the like terms:
That is the product rule for [cos(j) * cos(k)]. Now suppose I do [cos(j - k) - cos(j + k)]:
If we cancel out the like terms:
That is the product rule for [sin(j) * sin(k)].