To find the value of [tex]\( 1 - \cos(6x) \)[/tex], let's use trigonometric identities to simplify the expression.
Recall the double-angle identity for cosine:
[tex]\[
\cos(2\theta) = 1 - 2\sin^2(\theta)
\][/tex]
We want to transform [tex]\( 1 - \cos(6x) \)[/tex] into a form involving sine. Notice that [tex]\( \cos(6x) \)[/tex] can be written in terms of sine using the identity for double angles.
Let's start by setting up our given expression and look for a way to apply the trigonometric identities:
[tex]\[
1 - \cos(6x)
\][/tex]
We know that:
[tex]\[
1 - \cos(2\theta) = 2\sin^2(\theta)
\][/tex]
In our case, we are dealing with [tex]\( 6x \)[/tex] instead of [tex]\( 2\theta \)[/tex]. Therefore, let’s substitute [tex]\( \theta = 3x \)[/tex] so that [tex]\( 2\theta = 6x \)[/tex].
Now, substituting:
[tex]\[
1 - \cos(6x) = 1 - \cos(2 \times 3x)
\][/tex]
Using our identity:
[tex]\[
1 - \cos(2A) = 2\sin^2(A)
\][/tex]
Here, [tex]\( 2A = 6x \)[/tex] implies [tex]\( A = 3x \)[/tex]. Thus:
[tex]\[
1 - \cos(6x) = 2\sin^2(3x)
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
1 - \cos(6x) = 2\sin^2(3x)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{2 \sin ^2(3 x)}
\][/tex]
This corresponds to option C.
