# Another problem

## Homework Statement

sin5xcos3x=sin4xcos4x+sinxcosx, solve the identity

## Homework Equations

all the identities and formulas mentioned in my last thread.

## The Attempt at a Solution

Alright so I thought I could use the product to sum formula on the left side which ended up being 1/2sin10x then I used the sum/difference formula on the right side and got sin5x. Now according to my calculator these are not the same, so where have I gone wrong?

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Defennder
Homework Helper
Applying the product to sum formula to the left side is correct. But you don't end up with sin(10x). Where did you get that from?

original problem sin5xcos3x=sin4xcos4x+sinxcosx

Left side sin5xcos3x

Product to sum formula sinxcosx=1/2[sin(x+y)+sin(x-y)]
plug in the numbers

1/2[sin(5x+3x)+sin(5x-3x)]
simplify
1/2(sin8x+sin2x)
simplify
1/2sin10x

right side sin4xcos4x+sinxcosx

sum/difference formula sin(x+y) = sinxcosy + cosxsiny
plug in the numbers
oh,never mind that wont work its in the form sinxcosx +sinycosy hmmm.

dynamicsolo
Homework Helper
You're more than halfway home. You have on the left side

1/2[sin(5x+3x)+sin(5x-3x)]
simplify
1/2(sin8x+sin2x)
This is fine. You can leave the left side now.

1/2(sin8x+sin2x)
simplify
1/2sin10x
Alas, angles don't add this way...

right side sin4xcos4x+sinxcosx
Here, don't apply the product-sum identity to the second term (well, you could, but it's overly fussy). What is sin x·cos x equal to (using another identity)? That will get you one of the pieces you're after.

Do use the product-sum identity to the first term -- now you'll be done...

Thank you so much!