To rewrite the given expression [tex]\(\cos(7y) \cos(3y) - \sin(7y) \sin(3y)\)[/tex] as a single trigonometric function, we can use the trigonometric identities provided. Specifically, we will use the cosine addition identity.
The cosine addition identity states:
[tex]\[
\cos(A \pm B) = \cos(A) \cos(B) \mp \sin(A) \sin(B)
\][/tex]
In this identity:
- The plus sign in the argument of the cosine function ([tex]\(\pm\)[/tex]) corresponds to the minus sign between the terms in the expression ([tex]\(\mp\)[/tex]).
Given the expression:
[tex]\[
\cos(7y) \cos(3y) - \sin(7y) \sin(3y)
\][/tex]
We need to match this with one of the identities. Specifically, we notice that it fits the form of:
[tex]\[
\cos(A + B) = \cos(A) \cos(B) - \sin(A) \sin(B)
\][/tex]
Here, we can see that:
- [tex]\(A = 7y\)[/tex]
- [tex]\(B = 3y\)[/tex]
Therefore, the given expression can be rewritten as:
[tex]\[
\cos(7y + 3y)
\][/tex]
Simplifying inside the cosine function:
[tex]\[
\cos(10y)
\][/tex]
So, the expression [tex]\(\cos(7y) \cos(3y) - \sin(7y) \sin(3y)\)[/tex] is equivalent to:
[tex]\[
\cos(10y)
\][/tex]
Thus, the answer is:
[tex]\[
\cos(10y)
\][/tex]
