SOLUTION: cos3xcos5x-cos7xcos9x=0

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Question 891903: cos3xcos5x-cos7xcos9x=0
Answer by Edwin McCravy(20081) About Me  (Show Source):
You can put this solution on YOUR website!
cos%283x%29cos%285x%29-cos%287x%29cos%289x%29%22%22=%22%22%220%22

We use the product to sum formula on each term

cos%28alpha%29cos%28beta%29%22%22=%22%22expr%281%2F2%29%28cos%28alpha%2Bbeta%29%2Bcos%28alpha-beta%29%29

Swap the factors in each product so the difference will be positive:

cos%285x%29cos%283x%29-cos%289x%29cos%287x%29%22%22=%22%22%220%22 

expr%281%2F2%29%28cos%285x%2B3x%29%2Bcos%285x-3x%29%29%22%22-%22%22expr%281%2F2%29%28cos%289x%2B7x%29%2Bcos%289x-7x%29%29%22%22=%22%22%220%22

expr%281%2F2%29%28cos%288x%29%2Bcos%282x%29%29%22%22-%22%22expr%281%2F2%29%28cos%2816x%29%2Bcos%282x%29%29%22%22=%22%22%220%22

%28cos%288x%29%2Bcos%282x%29%29-%28cos%2816x%29%2Bcos%282x%29%29%22%22=%22%22%220%22

cos%288x%29%2Bcos%282x%29-cos%2816x%29-cos%282x%29%22%22=%22%22%220%22

cos%288x%29%2Bcross%28cos%282x%29%29-cos%2816x%29-cross%28cos%282x%29%29%22%22=%22%22%220%22

cos%288x%29-cos%2816x%29%22%22=%22%22%220%22

Use the doubla angle identity cos%282theta%29%22%22=%22%222cos%5E2%28theta%29-1
with theta=8x, 2theta=16x

cos%288x%29-%282cos%5E2%288x%29-1%29%22%22=%22%22%220%22

cos%288x%29-2cos%5E2%288x%29%2B1%22%22=%22%22%220%22

-2cos%5E2%288x%29%2Bcos%288x%29%2B1%22%22=%22%22%220%22

Multiply through by -1

2cos%5E2%288x%29-cos%288x%29-1%22%22=%22%22%220%22

Factor the left side:

%282cos%288x%29%2B1%29%28cos%288x%29-1%29%22%22=%22%22%220%22

Use zero-factor property:

2cos%288x%29%2B1=0,     cos%288x%29-1=0

2cos%288x%29=-1,     cos%288x%29=1

cos%288x%29=-1%2F2,     8x=2n%2Api
                    
8x=+pi+%2B-+pi%2F3%2B2n%2Api,     x=expr%28n%2F4%29%2Api       
 
8x+=+%281+%2B-+1%2F3+%2B+2n%29pi

8x=+%283%2F3+%2B-+1%2F3+%2B6n%2F3%29pi

8x+=+%28%283+%2B-+1%2B6n%29%2F3%29pi

8x+=+%28%286n+%2B3+%2B-+1%29%2F3%29pi

x+=+%28%286n+%2B3+%2B-+1%29%2F24%29pi,     x=expr%28n%2F4%29%2Api

Edwin