SOLUTION: Write cos(4𝑥) cos(3𝑥) − sin(4𝑥) sin(3𝑥) as a single trig function.

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Question 1162367: Write cos(4𝑥) cos(3𝑥) − sin(4𝑥) sin(3𝑥) as a single trig function.
Found 2 solutions by math_helper, ikleyn:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
+e%5E%28i4x%29+=+cos%284x%29%2Bisin%284x%29+
+e%5E%28i3x%29+=+cos%283x%29%2Bisin%283x%29+

Solve for cos():
++cos%284x%29+=+e%5E%28i4x%29-isin%284x%29+
++cos%283x%29+=+e%5E%28i3x%29-isin%283x%29+

Now (ignoring sin(4x)sin(3x) for the time being):



Which can be "simplified" to:


and further expanded to:







++cos%284x%29cos%283x%29+-+sin%284x%29sin%283x%29+=+cos%287x%29+%2B+i%28expression%29+

The part called "expression" must be zero since the left hand side is real, therefore:
++cos%284x%29cos%283x%29+-+sin%284x%29sin%283x%29+=+cos%287x%29+


EDIT: Fixed cos(4x) cut & paste error, it now reads properly

Answer by ikleyn(52818) About Me  (Show Source):
You can put this solution on YOUR website!
.

Use the basic argument addition formula for cosine 


    cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b) 


and you momentarily get


    cos(4x)*cos(3x) - sin(4x)*sin(3x) = cos(4x + 3x) = cos(7x).    ANSWER

Solved.