Download Trigonometric Identities

Module 6 Lesson 4
Using Trigonometric Identities – Part 3
Simplifying Trigonometric Expressions
Trigonometric Identities are found in many different areas of mathematics and science.
In addition to what we’ve been doing, there are two more main uses of trig identities.
The first use is to simplify trigonometric expressions.
Consider that there’s a science experiment in progress. The scientist has found an
expression to go along with some data that he’s collected. The expression looks like
this:
sec  cos2 
cot 
Looks a little complicated…and time consuming. If only there were some way to write
that in simpler terms. Fortunately for us, there is.
We’ve talked about Pythagorean, Reciprocal and Quotient Identities, among others. We
can make substitutions using these identities which will make good things happen for
us.
Remember this Reciprocal Identity?
sec  
1
cos 
Remember this Quotient Identity?
cot  
cos 
sin 
Let’s substitute those into our original expression:
sec  cos2 
cot 
1
cos2 
becomes cos 
cos 
sin 
We can now use some basic Algebra to simplify. Notice on the top that one of the
cosines cancels out:
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1
cos2 
cos 
cos 
sin 
Now we divide cos by
becomes
cos 
cos 
sin 
cos 
. To do this, we multiply by the reciprocal:
sin 
cos 
cos 
sin 
becomes cos  
sin 
cos 
The cosines cancel and we’re left with sin  .
What did we just show? We showed that:
sec  cos2 
 sin 
cot 
Now if you were the scientist doing this experiment, which would you rather write out
in your notes hundreds of times?
You can double check your result by plugging some different values in for theta. After
all, if these two things are actually equal, you should be able to plug in any value at all
for theta and get a valid equation.
So select a random number. I’m going to choose 16 degrees.
Evaluate this:
sec(16) cos2 (16)
cot(16)
Is it the same as:
sin(16)
It is. (If you don’t think it is, you typed something in incorrectly).
How do you know what to do?
Sure this simplification is simple if your instructor tells you how to do them. But how do
you make some of these decisions for yourself?
Although I can’t anticipate and help you solve every problem, I can provide some
guidelines and tips for you to keep in mind as you simplify these trigonometric
expressions.
These tips aren’t in any order. Sometimes a tip will help in one problem and make
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things worse in another. These are just some different suggestions that are intended to
give you a starting point to simplify.
TIP #1: Try to write everything in terms of sine and cosine
The reason behind this is simple. TAN, COT, CSC, and SEC can all be written in
terms of sine and cosine. If everything in the problem says SIN or COS, you
might get some things to cancel out. In fact, that’s what happened in our
example.
TIP #2: If you see a ‘1’ or a squared term – think Pythagorean Identities
Countless times in these identities, you’ll run into a substitution involving one of
our 3 Pythagorean Identities or some variation thereof. You should be aware of
the 3 main identities and also how you might cleverly rewrite them.
For example, you know our first Pythagorean Identity is:
sin 2   cos2   1
But do you also realize you could write that as either of these:
cos2   1  sin 2 
sin 2   1  cos2 
Or even possibly:
 cos2   sin 2   1
 sin 2   cos2   1
Be aware. SEC and TAN go together. CSC and COT go together.
TIP #3: Use Basic Algebra!
There will be many times when you add/subtract fractions, factor,
multiply/divide (FOIL!), find a common denominator, combining like terms,
squaring, etc. etc. Keep an open mind.
TIP #4: There’s probably more than one way to simplify
There are going to be problems that will have more than one way to simplify.
There’s no ‘right’ or ‘wrong’ way to simplify, as long as at the end, you have the
simplest form.
TIP #5: The time factor
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If you’re spending more than 5-10 minutes on any one problem, you’re probably
either doing it wrong, or you just don’t know how to do it. Don’t be afraid to
stop and ask. These problems aren’t intended to be terrible and timeconsuming. There’s no reason to spend more than 5-10 minutes on a problem.
It just causes bad feelings and unnecessary frustration.
The best way to explain these problems is actually just to illustrate through some
examples. I’ll try to include as much explanation as I can, although I’ll start abbreviating
steps after a point.
Example 1:
Simplify:
sin 2 
sec 2   1
Here, we see a ‘1’ and a trig function squared. This screams Pythagorean Identity.
Locate the identity that involves secant and 1:
1  tan 2  sec 2 
Doing a little shuffling gives us:
tan 2  sec 2   1
We can now make the substitution in the original expression:
We rewrite tangent in terms of sine and cosine:
sin 2 
tan 2 
sin 2 
sin 2 
cos2 
sin 2  sin 2  cos2 

 2  cos2 
Use basic algebra to simplify:
2
sin 
1
sin 
2
cos 
And we’re done. We’ve simplified
sin 2 
into cos2 
2
sec   1
Example 2:
sin  cot  cos
Simplify:
Here, we’ll start by writing cotangent in terms of sine and cosine:
sin  
cos 
cos 
sin 
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Some basic algebra gives us:
sin  cos2 

1
sin 
You can see that I’ve rewritten sine as a fraction over 1. This is because I see that am
supposed to be adding these two parts together. We do that with a common
denominator. In this case, the common denominator is sin  . So I’ll need to multiply
sin 
the first fraction by
:
sin 
sin  cos2 

1
sin 
sin  sin  cos2 
(
)

sin  1
sin 
2
2
sin  cos 

sin 
sin 
Adding the fractions gives:
sin 2   cos2 
sin 
Another convenient Pythagorean Identity Substitution:
which simplifies to:
csc
So our end result:
sin  cot  cos simplifies to csc
Example 3:
Simplify:
1
sin 
(sec x  tan x )(1  sin x )
Start with some basic algebra (specifically FOIL):
(sec x  tan x)(1  sin x)
sec x  sec x sin x  tan x  tan x sin x
Put everything in terms of sine and cosine:
sec x  sec x sin x  tan x  tan x sin x
1
1
sin x
sin x
(
)(sin x ) 
(
)(sin x )
cos x cos x
cos x cos x
Order of operations tells us to multiply:
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1
1
sin x
sin x
(
)(sin x ) 
(
)(sin x )
cos x cos x
cos x cos x
1
sin x sin x
sin 2 x


(
)
cos x cos x cos x
cos x
We now have a common denominator, so let’s combine all the fractions. Conveniently,
the ‘sin x’ terms cancel out:
1
sin x sin x
sin 2 x


(
)
cos x cos x cos x
cos x
1  sin 2 x
cos x
1  sin 2 x cos2 x

 cos x
cos x
cos x
Pythagorean and simplify:
So we just showed that:
(sec x  tan x )(1  sin x ) simplifies to cos x
Verifying Trigonometric Identities:
What we’ve done so far is to look at expressions and simplify them. The other main part
of this process is verifying trig identities. This is what happens when instead of an
expression, you’re presented with an equation:
cos x(tan 2 x  1)  sec x
To verify this identity, we need to show that the two sides of the equation are equal.
We do that in the same way we did the problems above. We try to simplify and
manipulate the left side until it looks exactly like the right side.
When we verify identities, we generally try to change one side only during the process.
It might be tempting to involve the other side in some fashion, but try not to. Try to
exclusively work on one side (either left or right). Will it always be possible to only work
with one side? Probably not. But try.
cos x(tan 2 x  1)  sec x
This example illustrates one other point. How would you start this problem? There are
two distinct ways – either of which will eventually lead to the correct answer. Some of
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you might have started by using a Pythagorean Identity substitution. Some of you
might have thought “Let’s get rid of the parenthesis using the distributive property”.
Both are valid. Both will work. This just goes to show you that there will be several
ways to verify these identities. There isn’t usually a “right” and “wrong” way. There
might be a way that’s quicker, but as long as you verify what you’re supposed to, it
shouldn’t matter how you get there.
Here are the solutions. I think you’ll agree that both work, but one is quicker. I’ll leave
the reasoning to you. Ask if you don’t understand.
Example 4:
Verify this trigonometric identity: cos x(tan 2 x  1)  sec x
cos x(tan 2 x  1)  sec x
Here is the first method:
cos x(sec 2 x )  sec x
1
cos x( 2 )  sec x
cos x
1
 sec x
cos x
sec x  sec x
Nice and quick.
Here is the other way. Wow.
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cos x (tan 2 x  1)  sec x
(cos x )(tan 2 x )  cos x  sec x
sin 2 x
(cos x )( 2 )  cos x  sec x
cos x
(cos x )(sin 2 x ) cos x

 sec x
cos2 x
1
(cos x )(sin 2 x ) cos x cos 2 x

(
)  sec x
cos2 x
1 cos2 x
(cos x )(sin 2 x ) cos3 x

 sec x
cos2 x
cos2 x
(cos x )(sin 2 x )  cos3 x
 sec x
cos2 x
sin 2 x  cos2 x
 sec x
cos x
1
 sec x
cos x
sec x  sec x
Let’s do one more.
Example 5:
Verify this trigonometric identity:
cot 2 x
 1  sin 2 x
2
csc x
Start with some Pythagorean action:
Sines and cosines:
cos2 x
sin 2 x  1  sin 2 x
1
sin 2 x
Using Algebra on the left side, we can see:
So now we have:
cot 2 x
 1  sin 2 x
2
1  cot x
cos2 x
2
2
sin 2 x  ( cos x )( sin x )  cos2 x
1
sin 2 x
1
2
sin x
cos2 x  1  sin 2 x
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One final Pythagorean substitution:
1  sin 2 x  1  sin 2 x
Keep in mind that there’s no set way to do these processes. There are guidelines and
tips to help you, but there’s no set way to simplify/verify. There might be different ways
to solve problems, some quicker, some slower.
Be patient and persistent with these problems. As you do more and more of them,
they’ll get easier.
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