To simplify the expression sin(9x) cos(x) cos(9x) sin(x), we can use some trigonometric identities. Specifically, we can leverage the product-to-sum identities.
Firstly, recall that the product-to-sum identities allow us to express products of sine and cosine functions as sums. In this case, we will start with the product cos(x) sin(x). Using the identity:
sin(a) cos(b) = 0.5 [sin(a + b) + sin(a - b)]
We can express cos(x) sin(x) as:
cos(x) sin(x) = 0.5 [sin(2x)].
Now, substitute this back into our original expression:
sin(9x) * 0.5 [sin(2x)] * cos(9x).
This can be rewritten as:
0.5 sin(9x) cos(9x) sin(2x).
Next, we can also use the product-to-sum identity on the terms sin(9x) cos(9x):
sin(a) cos(b) = 0.5 [sin(a + b) + sin(a - b)]
Thus, we have:
sin(9x) cos(9x) = 0.5 [sin(18x)].
Now, we can combine everything together:
0.5 * 0.5 [sin(18x)] * sin(2x) = 0.25 sin(18x) sin(2x).
Finally, we can write this product in terms of sine, cosine, or tangent, which maintains the original relationships between the angles. Therefore, the expression sin(9x) cos(x) cos(9x) sin(x) can be succinctly represented by:
Final Expression: 0.25 sin(18x) sin(2x).