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Which of the following is equal to the square root of one minus the cos of two π₯?
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(A) The absolute value of the sin of π₯.
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(B) Two times the absolute value of the cos of π₯.
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(C) The square root of two times the absolute value of the cos of π₯.
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(D) Two times the absolute value of the sin of π₯.
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(E) The square root of two times the absolute value of the sin of π₯.
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Okay, we want to see about converting this given expression to one of the five of our answer options.
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Thinking along those lines, the first thing we can notice is that weβre taking the cos of two times some angle π₯.
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This suggests we make use of the double-angle identity of the cosine function.
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And in fact, there are three different forms that this identity takes.
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We can choose any of them.
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But notice that if we choose this third one, then upon making that substitution for cos of two π₯ under our square root, we would have negative one being added to positive one adding up to zero.
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This would simplify the expression under the square root.
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So letβs indeed choose this third form of the double-angle identity.
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When we make this substitution, indeed we find out that this negative one added to a positive one gives us zero.
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And multiplying all the sines through, we get the square root of two times the sin squared of π₯.
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This equals the square root of two times the square root of the sin squared of π₯.
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And here we have to be careful because we might be tempted to say that the square root of the sin squared of π₯ equals simply the sin of π₯.
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Note though that while the square root of the sin squared of π₯ would never be negative, sin π₯ by itself could be.
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As we simplify this expression then, weβll want to include absolute value bars around the sin of π₯.
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This ensures that no matter what the value of π₯, weβll never get a negative overall result.