To simplify the expression [tex]\(\cos^4(12\theta) - \sin^4(12\theta)\)[/tex] and show that it is equivalent to [tex]\(\cos(24\theta)\)[/tex], we can follow these steps:
1. Identify Trigonometric Identities and Patterns:
We have the expression [tex]\(\cos^4(12\theta) - \sin^4(12\theta)\)[/tex]. Notice that this can be written as a difference of squares:
[tex]\[
\cos^4(12\theta) - \sin^4(12\theta) = (\cos^2(12\theta))^2 - (\sin^2(12\theta))^2.
\][/tex]
2. Apply the Difference of Squares Formula:
Recall the difference of squares formula: [tex]\(a^2 - b^2 = (a + b)(a - b)\)[/tex]. Here, [tex]\(a = \cos^2(12\theta)\)[/tex] and [tex]\(b = \sin^2(12\theta)\)[/tex]. Applying this, we get:
[tex]\[
(\cos^2(12\theta))^2 - (\sin^2(12\theta))^2 = (\cos^2(12\theta) + \sin^2(12\theta))(\cos^2(12\theta) - \sin^2(12\theta)).
\][/tex]
3. Simplify Using Known Identities:
We know from the Pythagorean identity that:
[tex]\[
\cos^2(12\theta) + \sin^2(12\theta) = 1.
\][/tex]
Substituting this into our expression, we get:
[tex]\[
1 \cdot (\cos^2(12\theta) - \sin^2(12\theta)) = \cos^2(12\theta) - \sin^2(12\theta).
\][/tex]
4. Use the Double Angle Formula:
The last step involves recognizing that [tex]\(\cos^2(12\theta) - \sin^2(12\theta)\)[/tex] is a known trigonometric identity for the cosine of a double angle. Specifically,
[tex]\[
\cos(2A) = \cos^2(A) - \sin^2(A).
\][/tex]
In our case, [tex]\(A\)[/tex] is [tex]\(12\theta\)[/tex]. Hence,
[tex]\[
\cos^2(12\theta) - \sin^2(12\theta) = \cos(24\theta).
\][/tex]
Therefore, we have shown that:
[tex]\[
\cos^4(12\theta) - \sin^4(12\theta) = \cos(24\theta).
\][/tex]
This completes the proof and shows the equivalence as required.
