### Video Transcript

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

Okay, so here weβre evaluating this
expression. And we want to see which of these
five answer options it equals. The first thing we can notice is
that weβre taking the sin of two times some angle π₯. We can think of this then as the
sine of a double angle, where π₯ is that angle. Recalling the double-angle identity
for the sine function, we know that the sin of two π₯ equals two times the sin of π₯
times the cos of π₯. Making this substitution into our
square root gives us this result.

And now letβs consider this factor
of one. By the Pythagorean identity, this
unassuming number one is equal to the sine squared of an angle plus the cosine
squared of that same angle. Making that substitution gives us
this expression. So far, it seems as though weβre
complicating rather than simplifying the expression under our square root. But now that we have these three
terms β sin squared π₯, cos squared π₯, and two sin π₯ cos π₯ β say that we can
write them as the quantity cos π₯ minus sin π₯ squared. We see then that our squaring
operation and then taking the square root will effectively invert one another. So we might expect our final result
to be the cos of π₯ minus the sin of π₯.

Here though, we have to be careful
to make sure this result is truly equal to our original expression. When we think about the sine
function, we know it has a maximum of one and a minimum of negative one. This means if we think about it for
a moment, that one minus the sin of two π₯ will never be negative. Since this is true for our original
expression, it must also be true for our final one. However, itβs certainly possible
for the cos of π₯ minus the sin of π₯ to be negative. To correct this and make our
expression truly equal to the square root of one minus the sin of two π₯, weβll put
absolute value bars around it. And so this is our final
answer. Itβs the absolute value of the cos
of π₯ minus the sin of π₯ that equals the square root of one minus the sin of two
π₯.