>> This is your Math Gal,
Julie Harland.
Please visit my website
at YourMathGal.com where all
of my videos are organized
by topic.
On this video, we're going
to use the basic identities
to rewrite the expression
given and then simplify it.
And we're going to begin
by writing everything in terms
of sine and cosine.
[Pause] So,
these are the basic
identities,
the reciprocal identities,
tangent and cotangent,
in terms of sine and cosine
and the three Pythagorean
identities you need
to be familiar with in order
to work some
of these problems.
So, here's the first one.
1 over tangent squared x minus
1 over sine squared x.
So let's begin
by writing everything in terms
of sine and cosine
or we can use a reciprocal
identity to write tangent
squared x if that's
in the denominator.
Remember, you could write
that as cotangent squared x
in the numerator.
So, I'm going to do
that since it's
the reciprocal.
Then I have cotangent squared
x and the numerator
over 1 minus 1
over sine squared x. So,
I haven't written this
in terms of sine and cosine,
yet; and, in fact,
I'm just going to leave this
as cotangent squared x.
But this is one way to start.
And then we could change this
cotangent squared x to cosine
over sine.
So, if you have the cotangent
squared x,
it's really the cosine of x
over the sine of x squared.
So, basically,
each of them will be squared.
[Pause] All right.
So this is one way
to start this problem.
Now you have a common
denominator
which is convenient.
So, in the numerator I have
cosine squared x minus 1
over sine squared x. Now
at this point you may think
you're done.
But, cosine squared x minus 1
can be simplified a little
bit further.
So, you should be familiar
with the Pythagorean identity,
cosine squared x plus sine
squared x equals 1.
And so, if we could manipulate
this so that we could have
on one side
of the equation cosign squared
x minus 1, we'll be able
to do a substitution.
And in fact you can do that.
If you subtract 1
from both sides,
we have cosine squared x
minus 1.
If we subtract 1
from both sides,
we'll have that on that side
and then we'll have a negative
sine squared on the right.
This means I could replace the
numerator up here
with negative sine squared x.
All right?
So, that's one way
of getting a
different numerator.
Another way
of doing it is simply
replacing cosine squared x
with -- well, if I wanted
to solve for cosine squared x,
I'd have cosine squared x
equals 1 minus sine squared x.
So that's another way
to do it.
Let's just do it that way.
So, I'm going to replace this
with 1 minus sine squared x.
Now you want
to get really familiar
with these.
Cosine squared x can be
written as 1 minus sine
squared x. Sine squared x
could be written
as 1 minus cosine squared x.
So I've replaced it
but I also have this minus 1
over sine squared x.
But notice what happens.
1 minus 1 is going to go away
and you have negative sine
squared x. So,
if you would have done it the
first way I said,
just simply replace
that whole numerator
with negative sine squared x,
you would still end
up with negative sine squared
x in the numerator
and a sine squared x
in the denominator and now,
this means negative 1 times
that, right?
So, we can cancel the sine
squared x but you're going
to still have a 1
so your final answer is going
to be negative 1.
1. [Pause] Always keep
in mind, there's more
than one way to do all
of these problems.
Some people might notice
that -- they know
that cosine squared x is 1
minus sine squared.
Right? And that means the sine
squared x is 1 minus cosine
squared and when they see this
cosine squared x minus 1 right
here, they say, oh,
that's just the opposite
so instead
of it being sine squared x,
they figured out it's going
to be sine squared x. There's
all sorts of different ways
people think about this.
But in the end, we should end
up with that this whole thing
simplifies to negative 1.
[Pause] All right,
here's our next one.
I'm going to just begin
by rewriting the cotangent
in terms of cosine and sine.
So, this gives me cosine of x
over sine x times this sine x.
We still have minus 1.
And this all ready is in terms
of cosine so I just leave it
as is.
But when I do this,
I see I've got sine x
and sine x will cancel
so I just have cosine x minus
1 [Pause] times cosine
of x minus 1.
This is real similar
to the one we just did.
This is cosine squared x minus
1 when you --
because it's the difference
of two squares
when you do the foil method.
Cosine squared x minus 1.
Well, we just went over that
in the last video
that cosine squared x could be
written as 1 minus sine
squared x. Right?
So, this one, we just end
up with negative sine squared
x. It's easy
to leave your answer
as cosine squared x minus 1.
It's just slightly more
simplified maybe to think
of that as negative sine
squared x. The idea is
that these are actually all
equal to each other
at each line.
And it's not necessarily true
that negative sine squared x
is simplified to it,
but you should note
that they are really the same.
This is your Math Gal,
Julie Harland.
Please visit my website
at YourMathGal.com where all
of my videos are organized
by topic.