SOLUTION: Rewrite the expression as a simplified expression containing one term: cos (alpha + beta) cos beta + sin (alpha + beta) sin beta =

Question 246875: Rewrite the expression as a simplified expression containing one term:
cos (alpha + beta) cos beta + sin (alpha + beta) sin beta =
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
This is a fairly simple problem once one learns

The various formulas available to you, and
when and how to use them

It may just be a matter of memorization to remember this formula. But learning how and when to use it takes effort. The key to knowing when you can use it is to understand that the "x" and the "y" are no more than place holders. It doesn't really matter what they are. Maybe your Trig. textbook doesn't even use "x" and "y" for this formula. Maybe it uses "a" and "b" instead. It really doesn't matter. The "x" and "y" (or "a" and "b") are just placeholders for any expression

The formula we will use for your problem is:

To recognize that this is the formula to use, look for the pattern involved and not look for any specific variable names. Look at the left side of and see "the cos of a difference" and look at the right side and see "cos of something times cos of something else + sin of the same something as before times the sin of the same something else as before".

I hope you can see that your expression
matches the pattern of the right side of

where in place of the "x" we have "alpha + beta" and in place of the "y" we have "beta". This is how we know which formula we can use.

And since your expression matches the right side of cos(x-y), we can replace the whole expression with the left side of cos(x-y), using "alpha+beta" for "x" and "beta" for "y":

The beta's end up canceling out and we end up with:

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