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Chapter 5: Problem 33

Use a double-angle formula to rewrite the expression. $$4-8 \sin ^{2} x$$

Short Answer

Expert verified
The expression \(4-8 \sin ^{2} x\) can be rewritten as \(2\cos(2x)\) using the double-angle formula.

Step by step solution

01

Write Down the Expression

Start with the provided expression, which is \(4 - 8\sin^2(x)\). We are requested to rewrite this expression using the double-angle formula.
02

Recognize the Double-Angle Identity

Look for a double-angle formula that will fit the given expression. The identity \(\cos 2x = 1 - 2\sin^2 x\) appears to match well if \(1\) is replaced by \(4\), and \(-2\sin^2 x\) is replaced by \(-8\sin^2 x\).
03

Rewrite the Expression Using the Identity

Using the double-angle identity for cosine, the expression \(4 - 8\sin^2(x)\) can be rewritten as \(2\cos(2x)\). This is because, upon having \(4\) in place of \(1\), and \(-8\sin^2 x\) in place of \(-2\sin^2 x\) in the identity, it would equate to \(2\left(1-2\sin^{2}x\right)\), which simplifies to \(2\cos (2x)\).

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