In this discussion, we are going to prove the sum-to-product formulas for cosine using the sum and difference formulas for cosine.
I have already derived the sum and difference formulas in previously discussions, so it is assumed that you know how to use them before proving the sum-to-product formulas.
With that said, recall the sum and difference formulas for cosine. We know that:
cos(X+Y) = cosXcosY – sinXsinY
cos(X-Y)= cosXcosY + sinXsinY
Next, we will simply add the two formulas together to get:
cos(X+Y) + cos(X-Y) = 2cosXcosY
Let’s call this equation 1
Now, let
A= X+Y and
B=X-Y
Let’s call the above equations 2.
Add and subtract the two equations to get:
A + B = 2X and A-B = 2Y
Solve for X and Y to get:
(A+B)/2 = X and (A-B)/2 = Y
Let’s call the above equations 3.
Now plug equations 2 and 3 into equation 1 to get one of our sum-to-product formulas for sine.
cos(X+Y) + cos(X-Y) = 2cosXcosY
cosA + cosB = 2cos((A+B)/2)cos((A-B)/2)
For the other formula, we use the sum and difference formulas for cos again.
cos(X+Y) = cosXcosY – sinXsinY
cos(X-Y)= cosXcosY + sinXsinY
This time, we subtract the difference formula for cosine from the sum formula for cosine to get:
cos(X+Y) – cos(X-Y) =- 2sinXsinY
Let’s call this equation 4.
Now we plug equations 2 and 3 into 4 to get the other product-to-sum formula for cosine.
cosA – cosB = -2sin((A+B)/2)sin((A-B)/2)
And that is how you prove the product-to-sum-formulas for cosine!
Thank you reading/watching!
I hope that I can assist you on your math journey!
Sincerely,
➕MathAngel369➕
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