Sure, let's carefully apply the relevant trigonometric identities to simplify the given expression:
Given expression:
[tex]\[ \cos(7y) \cos(3y) - \sin(7y) \sin(3y) \][/tex]
We can use the cosine angle addition formula:
[tex]\[ \cos(A + B) = \cos(A) \cos(B) - \sin(A) \sin(B) \][/tex]
By comparing this formula with our given expression [tex]\(\cos(7y) \cos(3y) - \sin(7y) \sin(3y)\)[/tex], we see that it fits perfectly if we let [tex]\( A = 7y \)[/tex] and [tex]\( B = 3y \)[/tex]. So,
[tex]\[ \cos(7y) \cos(3y) - \sin(7y) \sin(3y) = \cos((7y) + (3y)) \][/tex]
Therefore, the given expression simplifies to:
[tex]\[ \cos(10y) \][/tex]
So, the simplified form of [tex]\(\cos(7y) \cos(3y) - \sin(7y) \sin(3y)\)[/tex] is:
[tex]\[ \cos(10y) \][/tex]