To simplify the expression [tex]\((1 - \cos \theta)(1 + \cos \theta)\)[/tex], we can follow these steps:
1. Recognize the Pattern: Notice that the expression [tex]\( (1 - \cos \theta)(1 + \cos \theta) \)[/tex] resembles the form of a difference of squares. The general formula for the difference of squares is:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
In our expression, let [tex]\( a = 1 \)[/tex] and [tex]\( b = \cos \theta \)[/tex].
2. Apply the Difference of Squares Formula: Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[
(1 - \cos \theta)(1 + \cos \theta) = 1^2 - (\cos \theta)^2
\][/tex]
3. Simplify the Result: Calculate the squares:
[tex]\[
1^2 = 1 \quad \text{and} \quad (\cos \theta)^2 = \cos^2 \theta
\][/tex]
So, we have:
[tex]\[
1^2 - (\cos \theta)^2 = 1 - \cos^2 \theta
\][/tex]
4. Use a Trigonometric Identity: Recall the Pythagorean trigonometric identity, which states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Rearrange this identity to express [tex]\( 1 - \cos^2 \theta \)[/tex] in terms of [tex]\( \sin^2 \theta \)[/tex]:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1 \implies \sin^2 \theta = 1 - \cos^2 \theta
\][/tex]
5. Substitute the Identity: Replace [tex]\( 1 - \cos^2 \theta \)[/tex] with [tex]\( \sin^2 \theta \)[/tex]:
[tex]\[
1 - \cos^2 \theta = \sin^2 \theta
\][/tex]
Therefore, the simplified form of [tex]\((1 - \cos \theta)(1 + \cos \theta)\)[/tex] is:
[tex]\[
\sin^2 \theta
\][/tex]
That's the final result!
