Question

Use a double-angle formula to rewrite the expression.
8−8(cosθ−sinθ)^2

Answer 1

To rewrite the expression \( 8 - 8(\cos \theta - \sin \theta)^2 \) using double-angle formulas, we start by simplifying \( (\cos \theta - \sin \theta)^2 \).

1. Expand \( (\cos \theta - \sin \theta)^2 \):
\[
(\cos \theta - \sin \theta)^2 = \cos^2 \theta - 2\cos \theta \sin \theta + \sin^2 \theta
\]
Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we can substitute this in:
\[
\cos^2 \theta + \sin^2 \theta = 1 \implies (\cos \theta - \sin \theta)^2 = 1 - 2\cos \theta \sin \theta
\]

2. Substitute this back into the original expression:
\[
8 - 8(\cos \theta - \sin \theta)^2 = 8 - 8(1 - 2\cos \theta \sin \theta)
\]
Simplifying inside the parentheses:
\[
= 8 - 8 + 16\cos \theta \sin \theta = 16\cos \theta \sin \theta
\]

3. Now, we can use the double-angle identity for sine, which states:
\[
\sin(2\theta) = 2\sin \theta \cos \theta \implies \cos \theta \sin \theta = \frac{1}{2} \sin(2\theta)
\]

4. Substitute this back into our expression:
\[
16\cos \theta \sin \theta = 16 \cdot \frac{1}{2} \sin(2\theta) = 8\sin(2\theta)
\]

Therefore, the expression \( 8 - 8(\cos \theta - \sin \theta)^2 \) can be rewritten as:

\[
\boxed{8\sin(2\theta)}
\]