>> Instructor: On this video,
we verify the following three trig identities.
We're going to be proving trig identities, and for the one
on this video, you'll need to know these identities,
the fundamental identities and the Pythagorean identities,
and you should know these well.
All right.
We want to verify this identity that you see here.
And a way to get started is you look and see
which side is more complicated.
What you want to do is manipulate one side,
using identities that you know,
until it looks like the other side.
So the left side is the more complicated side, so I'm going
to start by writing the left-hand side.
And what we want to do is eventually get it to look
like cotangent x, if possible.
So we look at this left-hand side and we think,
well, what could I do here?
Well, when I'm looking in the numerator,
I see that there is a factor of cosine x in each term.
So to me that seems like a natural thing to pull out,
to factor out, cosine x. The other thing I want you
to remember is you always have to write the argument,
which means the angle.
Don't leave off the x. Don't just write cosine, sine squared.
You need to write the whole thing.
So if I factor out the cosine x, I get sine squared x,
less cosine squared x -- and you know what,
I'm writing a little too big, so I'm going to make that smaller,
so I could fit it in --
plus cosine squared x. And that's the numerator.
And I have this sine x in the denominator.
Okay, now.
Do we see anything we could do from here?
Well, hopefully, you recognize sine squared x plus cosine
squared x as the Pythagorean identity, that's equal to 1.
So what I have is the cosine x times 1, over the sine x,
which is, of course, just cosine x over sine x. And check it out.
That's what we're trying to get to.
What is cosine over sine?
It's cotangent x. So there we are.
We just figured it out.
So there we go.
We just showed how starting with one side, we did something
to make it look like the other side.
Now if you wouldn't have factored out that cosine x,
and you tried something else,
you may have eventually gotten it to work.
I think, in this particular case,
that this is probably the fastest you could do it.
But there isn't just one way to prove an identity.
You just have to make sure that whatever you do at each step,
you are, you know, using the identities
that you already know.
All right.
We want to verify this identity, sine squared x
over cosine x equals secant of x minus cosine x.
So you usually want to try with a complicated side,
and I don't really know which one is more complicated, here.
Neither of them looks, you know, too complicated.
But do make sure you remember you can't cancel the x's here.
Right? So I notice
on the left-hand side I have a fraction,
but on the right-hand side I have a subtraction.
So I wonder if there's a way, by looking at what's
over on the left here, if I can get a subtraction in here,
and the sine squared x is my key.
Right? Remember that sine squared x plus cosine squared x
equals 1.
So sine squared x is 1 minus cosine squared x.
So that's what I'm going to do.
I'm going to start with this side, sine squared x
over cosine x. And how about if I rewrite
that as 1 minus cosine squared x. Now you have to look
over here and think, where am I trying to get?
Okay. I'm trying to get
to secant x minus cosine x. All right.
So let's see.
Now what could I do?
Well, now that I have a subtraction in the numerator
and just one single term in the denominator,
I could split that numerator up.
It's like a minus b, over c. You could just say 1 over cosine x,
minus cosine squared x over cosine x.
And let's see what happens here.
Well, what is 1 over cosine x?
Cool! Secant x, which is what I'm trying to get to.
Minus -- and what's going to happen over here?
Let's see.
The cosines are going to cancel.
Right? I'll have 1 left in the numerator.
Oops! I wrote sex.
Whoo! Didn't mean to write that.
Secant of x minus cosine x. I just verified it.
So I'm done.
Now you might wonder, well, what if you want
to start with the other side?
Well, let's go ahead and do that now.
What if I wanted to start with secant x minus cosine x and end
up with the other one.
So in other words, let's say I start off
with secant x minus cosine x, and I'm trying to make it look
like sine squared x over cosine x. Okay?
So -- well, what I usually do, if I don't have things in terms
of cosine and sine, that's the first thing I do,
is I change everything to sines and cosines.
So secant of x can be written as 1 over the cosine of x.
So I have 1 over the cosine of x, minus cosine x. Hmm.
Well, I've got a fraction minus something that's not a fraction,
so it seems natural to me to get a common denominator.
So let's multiply the top and bottom by cosine of x
over cosine of x. And that'll give me in the numerator, then,
1 minus cosine squared x over the cosine of x.
And then hopefully you'll recognize this is an identity
up in the numerator, again.
1 minus cosine squared x is sine squared x over cosine x.
And that's, I believe, exactly what we're trying to prove.
Let's go up here.
Was I trying to prove sine squared x over cosine x?
Yes, there it is.
Sine squared x over cosine x.
So somebody else might have done their proof this way.
There isn't just one way to do it.
He might start on the left side.
He might start on the right side.
The goal is to use identities until you get it
to look like the other side.
And you have to be good at your algebra, and you really have
to know your trig identities.
All right, here's the next identity,
tangent theta cosine theta plus cosecant sine squared theta is
equal to 2 sine theta.
So I always like to start on the more complicated side,
which is the left side.
To make this simple, what I'm going to do is just put a line
down here, so I'm going to say what the left side --
I'm just going to manipulate the left side.
A lot of time that's easier, and it doesn't take
as much space on your paper.
Okay. So to begin, I think a good strategy is change
everything in terms of sines and cosines.
So we'll take this tangent of theta and rewrite it
as sine theta over cosine theta, then times the cosine theta.
Plus -- all right, what's cosecant theta?
Well, that's 1 over sine theta, times,
and I have a sine squared theta.
Okay. All right.
Now what? Well, hey,
I can cancel these cosine thetas, to get sine theta.
And over here, at a sine squared theta over sine of theta,
so one of the sines is going to cancel, and I have a sine theta.
Well, that's like y plus y. That would be 2y.
So sine theta plus sine theta would be 2 sine theta,
and notice I manipulated it to look like the other side.
So we've just verified it.
It's much easier to start with a more complicated side and get it
to look like the other side.
And the key here, really, is to put everything
in terms of cosines and sines.