WEBVTT
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Let's take a look at trigonometric identities. An identity is a type of equation where f of x equals g of x
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for every value of x for which both functions are defined. Look at these examples over here on the left.
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X plus 1 quantity squared equals x squared plus 2x plus 1. It doesn't matter what value of x you plug into it.
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The left side will always equal the right side. The same is true for this bottom equation. This is one you should be
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familiar with. It's one of the Pythagorean identities. Cosine squared x plus sine squared x equals 1 for every value
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of x. If you plug in say x equals pi over 6, cosine squared pi over 6 plus sine squared pi over 6 will equal 1.
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Again, any value you plug in here for x, if you take cosine squared x plus sine squared x, you will always get 1.
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It's true for all values of x. In contrast, look at these two examples. They're not true for all values of x.
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We can easily see that by just say plugging in x equals 0. We can see on the left that 3 does not equal 6. The
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same thing on the bottom equation, cosine of 0 squared is not equal to 1/2. These are not true all the time.
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An identity is true for all values of x in the domain. Now there are two important things that you need to be
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able to do to be successful in establishing identities. One of them is that you need to know the basic trig identities.
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For example, you need to know the quotient identities, the reciprocal identities, and the Pythagorean identities,
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and the other thing that you need to be able to do is to manipulate expressions using algebra. For example,
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you need to be able to factor, you need to be able to reduce, you need to be able to get a common denominator.
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Let's take a look at an example.
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Here we want to multiply and simplify this expression. Now you need to be careful because there's a negative sign
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in front of this product. So you need to make sure that you multiply, and then you take the opposite of that product.
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So let's begin. First we're going to multiply cosine theta plus sine theta times cosine theta plus sine theta.
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I'm going to put a bracket here in front of my product. So what do we get when we multiply? We're going to get
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cosine squared theta plus 2 cosine theta sine theta plus sine squared theta, all over
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sine theta cosine theta. Now look inside the bracket. What do you see? I see a cosine squared theta plus
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a sine squared theta. Again, knowing those basic identities is very important, or you won't be able to
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recognize that. Cosine squared theta plus sine squared theta is 1, so I'll replace that. Now I'm ready
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to subtract what's inside the bracket, and let's see what we get. The 1's are going to subtract out, 1 minus 1.
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You're going to have the opposite. You're going to have minus 2 cosine theta sine theta
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in the numerator over sine theta cosine theta in the denominator, and now you recognize that we've
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got common factors in the numerator and denominator. We can reduce by sine theta and cosine theta
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and end up with just a negative 2. Alright, let's look at another example.
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Here we want to multiply and simplify this expression. Well if we didn't have to multiply to simplify, of course,
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this is just equal to 1, so we could get this. But you're going to see after we multiply that we get a nice expression.
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Let's go ahead and start. So we want to multiply the numerator. We're actually not going to multiply it out.
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We're just going to leave it this product. Now what is cosine theta plus 1 times cosine theta minus 1?
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Of course, that's cosine squared theta minus 1. This should sort of be recognizable to you because you
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know one of the Pythagorean identities is cosine squared theta plus sine squared theta equals 1.
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Well, I want to know what's cosine squared theta minus 1. So of course if I subtract 1 from each side
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and subtract [sine squared theta], I'm going to get that cosine squared theta minus 1 equals negative
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sine squared theta. So I can replace the denominator with negative sine squared theta. Now let's take a
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look. We notice that we have a factor of sine theta in the numerator and a sine theta in the denominator.
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We can reduce, and so we're left with cosine theta minus 1 in the numerator over negative sine theta
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in the denominator. So again, many times we'll multiply by some expression of 1 to try to establish the
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identity and get the other side. Alright, let's take a look at establishing identities.
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Here we want to establish the identity tangent theta plus cotangent theta times cosine theta equals
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cosecant theta. Well when we establish an identity, it's important that we work with only one side at a
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time. In this example, I'm going to work with the left-hand side, so I'm going to rewrite that side.
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Tangent theta plus cotangent theta times cosine theta. Now in this example what I'm going to do,
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which is done many times, is I'm going to change everything to sines and cosines. Remember that tangent theta is
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sine theta over cosine theta. Cotangent theta, using the quotient identity, is cosine theta over sine theta.
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Now inside the parentheses, I want to add those terms together, so I need to get a common denominator,
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which of course is cosine theta sine theta, which means that I'm multiplying numerator and denominator
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here by a sine theta, so I get sine squared theta for the first term. Here I'm multiplying by a cosine theta,
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so I get cosine squared theta. Look at the numerator, sine squared theta plus cosine squared theta.
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That should be familiar to you. That equals 1. Now when we multiply we see that we can reduce by a
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factor of cosine theta. We get 1 over sine theta, and of course, 1 over sine theta, the reciprocal identity,
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the sine and cosecant are reciprocals of each other, so this equals the cosecant which, of course, is the
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right hand side. So your goal is always to take one side and do algebra to it to get to the other side.
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Let's look at another example.
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Establish the identity 1 minus secant theta over tangent theta minus tangent theta over 1 minus secant
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theta equals 2 cotangent theta. What I'm going to do is work on the left hand side, and we want to combine
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the terms. We want to subtract, so we have to combine the terms. We need to get a common denominator,
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which would be tangent theta times 1 minus secant theta, which means that I'm multiplying numerator
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and denominator here by 1 minus secant theta. So I have 1 minus secant theta times another 1 minus
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secant theta, so I get 1 minus secant theta quantity squared. On the second term I'm multiplying by
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tangent theta over tangent theta, so I get minus tangent squared theta. Now we want to square this term.
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Remember when you square that term you're going to end up with three terms. We're going to square
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1 minus secant theta, and we get 1 minus 2 secant theta plus secant squared theta minus tangent squared
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theta all over tangent theta times 1 minus secant theta. Now look at the numerator. Remember one of
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the Pythagorean identities says that 1 plus tangent squared theta equals secant squared theta. So what
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you see is secant squared theta minus tangent squared theta, so if I subtract tangent squared theta from
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both sides I'm going to get 1. So I can replace all of this with a 1. So let's see what we get. We get
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1 minus 2 secant theta plus 1 over tangent theta [times] 1 minus secant theta, and of course
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1 plus 1 equals 2. Now here in the numerator we've got a common factor of 2. We can factor that out.
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Now we've got a common factor, numerator and denominator, of 1 minus secant theta. We can reduce
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by 1 minus secant theta, and we end up with 2 over tangent theta, and remember the reciprocal of tangent
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is cotangent. So this is the same thing as 2 cotangent theta, which of course is the right-hand side. So again,
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what you can see, it's important to be able to square terms. It's important to be able to recognize identities.
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So the algebra and the trig are very important skills in establishing these identities. Let's look at another example.
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In this problem we want to establish the identity 1 minus cosine theta over 1 plus cosine theta equals
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parentheses cosecant theta minus cotangent theta, quantity squared. Now remember when establishing
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identities, the rule of thumb is to pick the more complicated side. Well in this case, it's pretty close so I'm
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going to choose the right-hand side. But you could start with the left hand side, and what you would want
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to do is to multiply by the conjugate of the denominator, which is 1 minus cosine theta. So you'd multiply
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numerator and denominator by 1 minus cosine theta. I'm going to start with the right-hand side,
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cosecant theta minus cotangent theta, the quantity squared. Now when you square that you get
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cosecant squared theta minus 2 cosecant theta cotangent theta plus cotangent squared theta.
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Now what I'm going to do at this point is I'm going to change everything to sines and cosines. Cosecant
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squared is 1 over sine squared. Cosecant again is 1 over sine, so I have 2 over sine theta. Cotangent theta
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is cosine theta over sine theta, and cotangent squared theta is cosine squared theta over sine squared theta.
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Now let's see what we have. Notice that each of the denominators is sine squared theta, which means we
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can combine the terms, combine the numerators, to get everything over sine squared theta. So we have
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1 minus 2 cosine theta plus cosine squared theta. Now look at the numerator. You should notice that
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that factors, and it factors into 1 minus cosine theta, quantity squared. Now looking at the other side,
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which is always my guide, I see cosines. Remember you can always write a sine squared theta in terms
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of cosine squared because sine squared is 1 minus cosine squared. So now we're almost there. The only
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thing we need to do is to factor the denominator. 1 minus cosine squared theta factors into 1 minus
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cosine theta times 1 plus cosine theta, and now we can see that we have a common factor of 1 minus
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cosine theta. We can reduce by a factor of 1 minus cosine theta, and we're left with 1 minus cosine theta
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in the numerator over 1 plus cosine theta in the denominator, which of course you can see is the
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left-hand side. That's always your goal, to get to the other side. Alright, so remember that sometimes you
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want to multiply by the conjugate of the denominator, which in this case would be 1 minus cosine theta
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over 1 minus cosine theta. The problem could have been done that way, and you would have basically
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just reversed these steps to get to the right-hand side. Alright, let's look at another example.
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In this problem we want to establish the identity sine theta minus secant theta over sine theta secant theta
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equals cosine theta minus cosecant theta. In this problem I'm going to start with the left-hand side and
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what I'm going to do, first let me just write this side. Many times what we want to do is combine the terms.
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In this problem what I want to do is separate the terms. So what I'm going to do is separate this into two
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terms. This is the same thing as sine theta over sine theta secant theta minus secant theta over sine theta
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secant theta because remember one of the common mistakes a student might make is to think that
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there's a common factor. But there's not a common factor to each of these three terms. Now when I separate it
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into these two terms, I notice that in this term I can reduce by a factor of sine theta to get 1 over secant theta,
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and here I can reduce by a factor of secant theta and get 1 over sine theta, and then knowing the
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reciprocal identities, 1 over secant is cosine theta, and 1 over sine is cosecant theta, which of course you
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can see is the right-hand side. Now as you can see from these examples, it's very important that you
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know the basic trig identities very well so that when you see an expression you know what to replace it
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with, like sine squared theta plus cosine squared theta equals 1. And you need to go through these
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examples, practice them until you know how to do them very well without any assistance.