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Chapter 5: Problem 64

Verify the identity. $$\sin ^{4} x+\cos ^{4} x=1-2 \cos ^{2} x+2 \cos ^{4} x$$

Short Answer

Expert verified
Yes, the given identity is true

Step by step solution

01

Express \(\sin^4{x}\) in terms of \(\cos^{4}{x}\)

Begin by expressing \(\sin^4{x}\) as \((1-\cos^2{x})^2\). This results in a purely cosine expression \((1 - \cos^2x)^2 + \cos^4x\).
02

Simplify the expression

Now expand the term \((1-\cos^2{x})^2\) to get \(1 - 2\cos^2x + \cos^4x\) and then add the \(\cos^4x\) term that we have from the initial equation. This results in \(1-2 \cos ^{2} x+2 \cos ^{4} x\).
03

Confirm the identity

Comparing the result with the right side of the original identity, it is seen that they are indeed the same. Hence, the identity is confirmed.

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