In this discussion, we are going to prove the product-to-sum formulas using the sum and difference formulas.
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 product-to-sum formulas.
Here is an outline of this discussion:
🔹 Proving sin(A)cos(B) = 1/2 [sin(A+B) + sin(A-B)]
🔹 Proving cos(A)sin(B) = 1/2[sin(A+B) -sin(A-B)]
🔹 Proving cos(A)cos(B) = 1/2[cos(A+B) + cos(A-B)]
🔹 Proving sin(A)sin(B) = 1/2[cos(A-B) – cos(A+B)]
Let’s begin!
🔹 Proving sin(A)cos(B) = 1/2[sin(A+B) + sin(A-B)]
Recall the sum and difference formulas for sine. We know that:
sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
sin(A-B) = sin(A)cos(B) – cos(A)sin(B)
Adding the two equations, we have
sin(A+B) + sin(A-B) = 2sin(A)cos(B)
Dividing both sides by two, we have,
[sin(A+B) + sin(A-B)]/2 = sin(A)cos(B)
Rewriting, we have,
sin(A)cos(B) = 1/2[sin(A+B) + sin(A-B)]
🔹 Proving cos(A)sin(B) = 1/2[sin(A+B) -sin(A-B)]
For the next product-to-sum formula, we subtract the difference formula for sine from the sum formula for sine.
sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
sin(A-B) = sin(A)cos(B) – cos(A)sin(B)
Subtracting, we have,
sin(A+B) – sin(A-B) = 2cos(A)sin(B)
Dividing both sides by two, we have,
[sin(A+B) – sin(A-B)]/2 = cos(A)sin(B)
Rewriting, we have,
cos(A)sin(B) = 1/2[sin(A+B) -sin(A-B)]
🔹 Proving cos(A)cos(B) = 1/2[cos(A+B) + cos(A-B)]
For the next product to sum formula, we use the sum and difference formulas for cosine. We know that:
cos(A+B) = cos(A)cos(B) – sin(A)sin(B)
cos(A-B) = cos(A)cos(B) + sin(A)sin(B)
Adding the two equations, we have
cos(A+B) + cos(A-B) = 2cos(A)cos(B)
Dividing both sides by 2, we have
[cos(A+B) + cos(A-B)]/2= cos(A)cos(B)
Rewriting, we have,
cos(A)cos(B) = 1/2[cos(A+B) + cos(A-B)]
🔹 Proving sin(A)sin(B) = 1/2[cos(A-B) – cos(A+B)]
For the last product-to-sum formula, we subtract the difference formula for cosine from the sum formula for cosine.
cos(A+B) = cos(A)cos(B) – sin(A)sin(B)
cos(A-B) = cos(A)cos(B) + sin(A)sin(B)
Subtracting, we have,
cos(A+B) – cos(A-B) = -2sin(A)sin(B)
Dividing both side by -2, we have,
[cos(A-B) – cos(A+B)]/2 = sin(A)sin(B)
Rewriting, we have,
sin(A)sin(B) = 1/2[cos(A-B) – cos(A+B)]
And that is how you prove the product-to-sum formulas!
Thank you reading/watching!
I hope that I can assist you on your math journey!
Sincerely,
➕MathAngel369➕
🎥 Suggested Video and Playlist:
Video: Proof of the Sum-to-Product Formulas for Sine
Video: Proof of the Product-to-Sum Formulas for Cosine
Video: Deriving the Sum and Difference Formulas/Identities for Sine and Cosine
Playlist: Proving/Verifying and Using Trig Identities
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