To express the product [tex]\(\sin(11x) \cos(x)\)[/tex] as a sum or difference of sines and/or cosines, we can make use of a specific trigonometric identity. The relevant identity is:
[tex]\[
\sin(A) \cos(B) = \frac{1}{2} \left[ \sin(A + B) + \sin(A - B) \right]
\][/tex]
Here, [tex]\(A = 11x\)[/tex] and [tex]\(B = x\)[/tex].
Using the identity mentioned, we proceed as follows:
1. Identify [tex]\(A + B\)[/tex]:
[tex]\[
A + B = 11x + x = 12x
\][/tex]
2. Identify [tex]\(A - B\)[/tex]:
[tex]\[
A - B = 11x - x = 10x
\][/tex]
3. Substitute [tex]\(A + B\)[/tex] and [tex]\(A - B\)[/tex] back into the identity:
[tex]\[
\sin(11x) \cos(x) = \frac{1}{2} \left[ \sin(12x) + \sin(10x) \right]
\][/tex]
Thus, the product [tex]\(\sin(11x) \cos(x)\)[/tex] can be written as:
[tex]\[
\sin(11x) \cos(x) = \frac{\sin(12x)}{2} + \frac{\sin(10x)}{2}
\][/tex]
So, the final simplified answer is:
[tex]\[
\sin(11x) \cos(x) = \frac{1}{2} \sin(12x) + \frac{1}{2} \sin(10x)
\][/tex]
