### Video Transcript

Which of the following is equal to
the square root of one minus the cos of two π₯? (A) The absolute value of the sin
of π₯. (B) Two times the absolute value of
the cos of π₯. (C) The square root of two times
the absolute value of the cos of π₯. (D) Two times the absolute value of
the sin of π₯. (E) The square root of two times
the absolute value of the sin of π₯.

Okay, we want to see about
converting this given expression to one of the five of our answer options. Thinking along those lines, the
first thing we can notice is that weβre taking the cos of two times some angle
π₯. This suggests we make use of the
double-angle identity of the cosine function. And in fact, there are three
different forms that this identity takes. We can choose any of them. 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. This would simplify the expression
under the square root. So letβs indeed choose this third
form of the double-angle identity.

When we make this substitution,
indeed we find out that this negative one added to a positive one gives us zero. And multiplying all the sines
through, we get the square root of two times the sin squared of π₯. This equals the square root of two
times the square root of the sin squared of π₯. 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₯. Note though that while the square
root of the sin squared of π₯ would never be negative, sin π₯ by itself could
be. As we simplify this expression
then, weβll want to include absolute value bars around the sin of π₯. This ensures that no matter what
the value of π₯, weβll never get a negative overall result.

Our final answer then is that itβs
the square root of two times the absolute value of the sin of π₯ thatβs equal to the
square root of one minus the cos of two π₯.