How do you use the product to sum formulas to write #5cos(-5beta)cos3beta# as a sum or difference?

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Answer 1 · 2017-02-17T14:53:49

See below

Firstly, the sum of cosines is always written in the form #2cos(("P"+"Q")/2)cos(("P"-"Q")/2)#, so we have to take the #5# out.

#5cos(-5beta)cos3beta=5/2(2cos(-5beta)cos3beta)#

Since #cosx# is an even function, we know that #cos(-5beta)=cos(5beta)#

#5/2(2cos(-5beta)(3beta))=5/2(2cos5betacos3beta)#

Now it is trivial to find #"P"# and #"Q"#

#"P"=5beta+3beta=8beta#

#"Q"=5beta-3beta=2beta#

#5/2(2cos5betacos3beta)=5/2(cos8beta+cos3beta)#