Question
Look at the proof showing sine superscript 4 baseline x equals start fraction 3 minus 4 cosine 2 x plus cosine 4 x over 8 end fraction. Which expression will complete the fourth step of the proof?
math expressionsine superscript 4 baseline x equals left parenthesis sine squared x right parenthesis squared line break
sine superscript 4 baseline x equals left parenthesis start fraction 1 minuscosine 2 x over 2 end fraction right parenthesis squared line break
sine superscript 4 baseline x equals start fraction 1 minus 2 cosine 2 x plus cosine squared 2 x over 4 end fraction line break
sine superscript 4 baseline x equals line break
sine superscript 4 baseline x equals start fraction 2 minus 4 cosine 2 x plus 1 plus cosine 4 x over 8 end fraction line break
sine superscript 4 baseline x equals start fraction 3 minus 4 cosine 2 x plus cosine 4 x over 8 end fraction
(1 point)
Responses
start fraction 1 minus 2 cosine 2 x plus start fraction 1 plus cosine 4 x over 2 end fraction over 4 end fraction
Image with alt text: start fraction 1 minus 2 cosine 2 x plus start fraction 1 plus cosine 4 x over 2 end fraction over 4 end fraction
start fraction 1 minus 2 cosine 2 x plus 1 plus cosine 4 x over 4 end fraction
Image with alt text: start fraction 1 minus 2 cosine 2 x plus 1 plus cosine 4 x over 4 end fraction
start fraction 1 minus 2 cosine 2 x plus start fraction 1 plus cosine 2 x over 2 end fraction over 4 end fraction
Image with alt text: start fraction 1 minus 2 cosine 2 x plus start fraction 1 plus cosine 2 x over 2 end fraction over 4 end fraction
start fraction 1 minus 2 cosine 2 x plus 1 minus cosine 2 x over 4 end fraction
Image with alt text: start fraction 1 minus 2 cosine 2 x plus 1 minus cosine 2 x over 4 end fraction
Skip to navigation
math expressionsine superscript 4 baseline x equals left parenthesis sine squared x right parenthesis squared line break
sine superscript 4 baseline x equals left parenthesis start fraction 1 minuscosine 2 x over 2 end fraction right parenthesis squared line break
sine superscript 4 baseline x equals start fraction 1 minus 2 cosine 2 x plus cosine squared 2 x over 4 end fraction line break
sine superscript 4 baseline x equals line break
sine superscript 4 baseline x equals start fraction 2 minus 4 cosine 2 x plus 1 plus cosine 4 x over 8 end fraction line break
sine superscript 4 baseline x equals start fraction 3 minus 4 cosine 2 x plus cosine 4 x over 8 end fraction
(1 point)
Responses
start fraction 1 minus 2 cosine 2 x plus start fraction 1 plus cosine 4 x over 2 end fraction over 4 end fraction
Image with alt text: start fraction 1 minus 2 cosine 2 x plus start fraction 1 plus cosine 4 x over 2 end fraction over 4 end fraction
start fraction 1 minus 2 cosine 2 x plus 1 plus cosine 4 x over 4 end fraction
Image with alt text: start fraction 1 minus 2 cosine 2 x plus 1 plus cosine 4 x over 4 end fraction
start fraction 1 minus 2 cosine 2 x plus start fraction 1 plus cosine 2 x over 2 end fraction over 4 end fraction
Image with alt text: start fraction 1 minus 2 cosine 2 x plus start fraction 1 plus cosine 2 x over 2 end fraction over 4 end fraction
start fraction 1 minus 2 cosine 2 x plus 1 minus cosine 2 x over 4 end fraction
Image with alt text: start fraction 1 minus 2 cosine 2 x plus 1 minus cosine 2 x over 4 end fraction
Skip to navigation
Answers
GPT-4o mini
To complete the proof that shows \(\sin^4 x = \frac{3 - 4 \cos 2x + \cos 4x}{8}\), we can break down the proof step by step.
1. Start with the identity \(\sin^2 x = \frac{1 - \cos 2x}{2}\).
2. Then square both sides:
\[
\sin^4 x = \left(\frac{1 - \cos 2x}{2}\right)^2 = \frac{(1 - \cos 2x)^2}{4}
\]
3. Expanding \((1 - \cos 2x)^2\) gives:
\[
\sin^4 x = \frac{1 - 2\cos 2x + \cos^2 2x}{4}
\]
4. Now, we can express \(\cos^2 2x\) in terms of \(\cos 4x\) using the identity \(\cos^2 \theta = \frac{1 + \cos 2\theta}{2}\). This gives us:
\[
\cos^2 2x = \frac{1 + \cos 4x}{2}
\]
5. Substitute \(\cos^2 2x\) into the expression from step 3:
\[
\sin^4 x = \frac{1 - 2\cos 2x + \frac{1 + \cos 4x}{2}}{4}
\]
To complete the fourth step of the proof, we need to combine like terms. This results in the expression:
\[
\sin^4 x = \frac{1 - 2\cos 2x + \frac{1 + \cos 4x}{2}}{4}
\]
Now we can rewrite this as:
\[
\sin^4 x = \frac{2 - 4\cos 2x + 1 + \cos 4x}{8}
\]
or simplified,
\[
\sin^4 x = \frac{3 - 4\cos 2x + \cos 4x}{8}
\]
The expression that will complete the fourth step before arriving at the final result is:
\[
\frac{1 - 2\cos 2x + \frac{1 + \cos 4x}{2}}{4}
\]
Looking at the provided options, the correct choice appears to be:
**start fraction 1 minus 2 cosine 2 x plus start fraction 1 plus cosine 4 x over 2 end fraction over 4 end fraction**
1. Start with the identity \(\sin^2 x = \frac{1 - \cos 2x}{2}\).
2. Then square both sides:
\[
\sin^4 x = \left(\frac{1 - \cos 2x}{2}\right)^2 = \frac{(1 - \cos 2x)^2}{4}
\]
3. Expanding \((1 - \cos 2x)^2\) gives:
\[
\sin^4 x = \frac{1 - 2\cos 2x + \cos^2 2x}{4}
\]
4. Now, we can express \(\cos^2 2x\) in terms of \(\cos 4x\) using the identity \(\cos^2 \theta = \frac{1 + \cos 2\theta}{2}\). This gives us:
\[
\cos^2 2x = \frac{1 + \cos 4x}{2}
\]
5. Substitute \(\cos^2 2x\) into the expression from step 3:
\[
\sin^4 x = \frac{1 - 2\cos 2x + \frac{1 + \cos 4x}{2}}{4}
\]
To complete the fourth step of the proof, we need to combine like terms. This results in the expression:
\[
\sin^4 x = \frac{1 - 2\cos 2x + \frac{1 + \cos 4x}{2}}{4}
\]
Now we can rewrite this as:
\[
\sin^4 x = \frac{2 - 4\cos 2x + 1 + \cos 4x}{8}
\]
or simplified,
\[
\sin^4 x = \frac{3 - 4\cos 2x + \cos 4x}{8}
\]
The expression that will complete the fourth step before arriving at the final result is:
\[
\frac{1 - 2\cos 2x + \frac{1 + \cos 4x}{2}}{4}
\]
Looking at the provided options, the correct choice appears to be:
**start fraction 1 minus 2 cosine 2 x plus start fraction 1 plus cosine 4 x over 2 end fraction over 4 end fraction**