To prove that [tex]\(\cos^4 \left( \frac{A}{2} \right) - \sin^4 \left( \frac{A}{2} \right) = \cos A\)[/tex], we can use algebraic transformations and trigonometric identities. Here's a step-by-step solution:
1. Recognize a Difference of Squares:
Notice that [tex]\(\cos^4 \left( \frac{A}{2} \right) - \sin^4 \left( \frac{A}{2} \right)\)[/tex] is a difference of squares:
[tex]\[
\cos^4 \left( \frac{A}{2} \right) - \sin^4 \left( \frac{A}{2} \right) = \left( \cos^2 \left( \frac{A}{2} \right) \right)^2 - \left( \sin^2 \left( \frac{A}{2} \right) \right)^2
\][/tex]
Now apply the difference of squares formula, [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]:
[tex]\[
\left( \cos^2 \left( \frac{A}{2} \right) \right)^2 - \left( \sin^2 \left( \frac{A}{2} \right) \right)^2 = \left( \cos^2 \left( \frac{A}{2} \right) - \sin^2 \left( \frac{A}{2} \right) \right)\left( \cos^2 \left( \frac{A}{2} \right) + \sin^2 \left( \frac{A}{2} \right) \right)
\][/tex]
2. Use the Pythagorean Identity:
The Pythagorean identity tells us that [tex]\(\cos^2 x + \sin^2 x = 1\)[/tex]. Applying this to our expression where [tex]\(x = \frac{A}{2}\)[/tex]:
[tex]\[
\cos^2 \left( \frac{A}{2} \right) + \sin^2 \left( \frac{A}{2} \right) = 1
\][/tex]
Therefore, our expression simplifies to:
[tex]\[
\left( \cos^2 \left( \frac{A}{2} \right) - \sin^2 \left( \frac{A}{2} \right) \right) \cdot 1 = \cos^2 \left( \frac{A}{2} \right) - \sin^2 \left( \frac{A}{2} \right)
\][/tex]
3. Use the Cosine Double-Angle Identity:
Recall the cosine double-angle identity: [tex]\(\cos (2x) = \cos^2 x - \sin^2 x\)[/tex].
For [tex]\(x = \frac{A}{2}\)[/tex], we have:
[tex]\[
\cos A = \cos \left( 2 \cdot \frac{A}{2} \right) = \cos^2 \left( \frac{A}{2} \right) - \sin^2 \left( \frac{A}{2} \right)
\][/tex]
4. Conclusion:
So, substituting back to our original problem, we get:
[tex]\[
\cos^4 \left( \frac{A}{2} \right) - \sin^4 \left( \frac{A}{2} \right) = \cos^2 \left( \frac{A}{2} \right) - \sin^2 \left( \frac{A}{2} \right) = \cos A
\][/tex]
Thus, we have proven that [tex]\(\cos^4 \left( \frac{A}{2} \right) - \sin^4 \left( \frac{A}{2} \right) = \cos A\)[/tex].
