### Video Transcript

Simplify one plus cot squared three
π over two minus π over one plus tan squared π over two minus π.

In order to answer this question,
we will need to use a variety of trigonometric identities. There are many ways to start
here. However, we will begin by trying to
rewrite the expression simply in terms of π. By firstly sketching the unit
circle, we recall that π radians is equal to 180 degrees. This means that π over two radians
is equal to 90 degrees. The denominator of our expression
can therefore be rewritten as one plus tan squared of 90 degrees minus π. One of our cofunction identities
states that tan of 90 degrees minus π is equal to cot π. This means that tan squared of 90
degrees minus π is equal to cot squared π. And the denominator of our
expression is therefore equal to one plus cot squared π.

Letβs now consider the angle three
π over two minus π. Once again, we can see from the
unit circle that three π over two radians is equal to 270 degrees. This means that the numerator of
our expression is equal to one plus cot of 270 degrees minus π. If π lies in the first quadrant,
as shown in our right triangle, then three π over two minus π, or 270 degrees
minus π, lies in the third quadrant. It is clear from the diagram that
cos of three π over two minus π is equal to negative sin π and sin of three π
over two minus π is equal to negative cos π. Since sin π over cos π is tan π
and cos π over sin π is cot π, then cot of 270 degrees minus π is equal to tan
π. Squaring both sides of this
identity, we can rewrite the numerator of our expression as one plus tan squared
π.

Our next step is to recall two of
the Pythagorean identities. Firstly, tan squared π plus one is
equal to sec squared π. And secondly, one plus cot squared
π is equal to csc squared π. Our expression simplifies to sec
squared π over csc squared π. And this can be rewritten as sec
squared π multiplied by one over csc squared π. Recalling the reciprocal identities
sec π is equal to one over cos π and csc π is equal to one over sin π, we have
one over cos squared π multiplied by sin squared π, which can be rewritten as sin
squared π over cos squared π and, in turn, is equal to tan squared π. The expression one plus cot squared
three π over two minus π over one plus tan squared π over two minus π written in
its simplest form is tan squared π.