Sure! Let's simplify the expression [tex]\((1 + \cos(x))(1 - \cos(x))\)[/tex] using trigonometric identities.
1. Recognize the trigonometric identity:
The expression [tex]\((1 + \cos(x))(1 - \cos(x))\)[/tex] can be recognized as a difference of squares. Recall the algebraic identity:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
In this case, [tex]\(a = 1\)[/tex] and [tex]\(b = \cos(x)\)[/tex].
2. Apply the difference of squares:
Applying the identity to our expression:
[tex]\[
(1 + \cos(x))(1 - \cos(x)) = 1^2 - \cos^2(x)
\][/tex]
Simplifying the right-hand side, we obtain:
[tex]\[
1 - \cos^2(x)
\][/tex]
3. Use the Pythagorean identity:
Recall the Pythagorean identity in trigonometry, which states:
[tex]\[
\sin^2(x) + \cos^2(x) = 1
\][/tex]
Solving for [tex]\(1 - \cos^2(x)\)[/tex], we get:
[tex]\[
1 - \cos^2(x) = \sin^2(x)
\][/tex]
4. Substitute back into the expression:
Now, replace [tex]\(1 - \cos^2(x)\)[/tex] with [tex]\(\sin^2(x)\)[/tex]:
[tex]\[
(1 + \cos(x))(1 - \cos(x)) = \sin^2(x)
\][/tex]
So, the simplified form of the expression [tex]\((1 + \cos(x))(1 - \cos(x))\)[/tex] is:
[tex]\[
\boxed{\sin^2(x)}
\][/tex]
