Question 984458
<pre>
cos(7<font face="symbol">q</font>) +cos(5<font face="symbol">q</font>) + cos(3<font face="symbol">q</font>) + cos(<font face="symbol">q</font>) = 4cos(<font face="symbol">q</font>)cos(2<font face="symbol">q</font>)cos(4<font face="symbol">q</font>)

The left is a sum and the right is a product. If we choose to work with the
left side we will need a sum-to-product cosine identity. If we choose to work
with the right side we will need a cosine product-to-sum identity. Since
products are more complicated than sums,  I will choose to work with the
right side. 

We need to derive a cosine product-to-sum identity:

Add the two well-known identities:

          cos(A+B) = cos(A)cos(B) - sin(A)sin(B)
          cos(A-B) = cos(A)cos(B) + sin(A)sin(B)
-------------------------------------------------
 cos(A+B)+cos(A-B) = 2cos(A)cos(B)

Divide both sides by 2:

[cos(A+B)+cos(A-B)]/2 = cos(A)cos(B)

Turn it around:

cos(A)cos(B) = [cos(A+B)+cos(A-B)]/2

This is the identity we will be using in addition to the
fact that cos(-A) = cos(A)

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Start with the right side:

4cos(<font face="symbol">q</font>)cos(2<font face="symbol">q</font>)cos(4<font face="symbol">q</font>)

          Use the identity to replace cos(<font face="symbol">q</font>)cos(2<font face="symbol">q</font>) by
          [cos(<font face="symbol">q</font>+2<font face="symbol">q</font>)+cos(<font face="symbol">q</font>-2<font face="symbol">q</font>)]/2 or [cos(3<font face="symbol">q</font>)+cos(-<font face="symbol">q</font>)]/2
          or [cos(3<font face="symbol">q</font>)+cos(<font face="symbol">q</font>)]/2

4[[cos(3<font face="symbol">q</font>)+cos(<font face="symbol">q</font>)]/2]cos(4<font face="symbol">q</font>)
2[cos(3<font face="symbol">q</font>)+cos(<font face="symbol">q</font>)]cos(4<font face="symbol">q</font>)
2cos(3<font face="symbol">q</font>)cos(4<font face="symbol">q</font>)+2cos(<font face="symbol">q</font>)cos(4<font face="symbol">q</font>)

          Use the identity to replace cos(3<font face="symbol">q</font>)cos(4<font face="symbol">q</font>) by
          [cos(3<font face="symbol">q</font>+4<font face="symbol">q</font>)+cos(3<font face="symbol">q</font>-4<font face="symbol">q</font>)]/2 or [cos(7<font face="symbol">q</font>)+cos(-<font face="symbol">q</font>)]/2
          or [cos(7<font face="symbol">q</font>)+cos(<font face="symbol">q</font>)]/2

          Also use the identity to replace cos(<font face="symbol">q</font>)cos(4<font face="symbol">q</font>) by
          [cos(<font face="symbol">q</font>+4<font face="symbol">q</font>)+cos(<font face="symbol">q</font>-4<font face="symbol">q</font>)]/2 or [cos(5<font face="symbol">q</font>)+cos(-3<font face="symbol">q</font>)]/2
          or [cos(5<font face="symbol">q</font>)+cos(3<font face="symbol">q</font>)]/2

2[cos(7<font face="symbol">q</font>)+cos(<font face="symbol">q</font>)]/2 + 2[cos(5<font face="symbol">q</font>)+cos(3<font face="symbol">q</font>)]/2

Cancel the 2's

cos(7<font face="symbol">q</font>)+cos(<font face="symbol">q</font>)+cos(5<font face="symbol">q</font>)+cos(3<font face="symbol">q</font>)

Swap the two middle terms:

cos(7<font face="symbol">q</font>)+cos(5<font face="symbol">q</font>)+cos(<font face="symbol">q</font>)+cos(3<font face="symbol">q</font>)

Swap the last two terms:

cos(7<font face="symbol">q</font>)+cos(5<font face="symbol">q</font>)+cos(3<font face="symbol">q</font>)+cos(<font face="symbol">q</font>)

Edwin</pre>