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Solving Trigonometric Equations
with Identities
∗
OpenStax College
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 4.0
Abstract
In this section, you will:
•
•
∗
†
Verify the fundamental trigonometric identities.
Simplify trigonometric expressions using algebra and the identities.
Version 1.5: Mar 11, 2015 10:47 am +0000
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†
OpenStax-CNX module: m49393
Figure 1:
2
International passports and travel documents
In espionage movies, we see international spies with multiple passports, each claiming a dierent identity.
However, we know that each of those passports represents the same person. The trigonometric identities
act in a similar manner to multiple passportsthere are many ways to represent the same trigonometric
expression. Just as a spy will choose an Italian passport when traveling to Italy, we choose the identity that
applies to the given scenario when solving a trigonometric equation.
In this section, we will begin an examination of the fundamental trigonometric identities, including how
we can verify them and how we can use them to simplify trigonometric expressions.
1 Verifying the Fundamental Trigonometric Identities
Identities enable us to simplify complicated expressions. They are the basic tools of trigonometry used in
solving trigonometric equations, just as factoring, nding common denominators, and using special formulas
are the basic tools of solving algebraic equations. In fact, we use algebraic techniques constantly to simplify
trigonometric expressions. Basic properties and formulas of algebra, such as the dierence of squares formula
and the perfect squares formula, will simplify the work involved with trigonometric expressions and equations.
We already know that all of the trigonometric functions are related because they all are dened in terms of
the unit circle. Consequently, any trigonometric identity can be written in many ways.
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To verify the trigonometric identities, we usually start with the more complicated side of the equation
and essentially rewrite the expression until it has been transformed into the same expression as the other side
of the equation. Sometimes we have to factor expressions, expand expressions, nd common denominators,
or use other algebraic strategies to obtain the desired result. In this rst section, we will work with the
fundamental identities: the Pythagorean identities, the even-odd identities, the reciprocal identities, and
the quotient identities.
We will begin with the Pythagorean identities (see Table 1), which are equations involving trigonometric functions based on the properties of a right triangle. We have already seen and used the rst of these
identies, but now we will also use additional identities.
Pythagorean Identities
2
2
sin θ + cos θ = 1
1 + cot2 θ = csc2 θ
1 + tan2 θ = sec2 θ
Table 1
The second and third identities can be obtained by manipulating the rst. The identity 1+cot θ = csc θ is
found by rewriting the left side of the equation in terms of sine and cosine.
Prove: 1 + cot θ = csc θ
1 + cot θ = 1 +
Rewrite the left side.
+
Write both terms with the common denominator.
=
(1)
=
2
2
2
2
cos2 θ
sin2 θ
2
sin2 θ
sin2 θ
=
cos2 θ
sin2 θ
sin2 θ+cos2 θ
sin2 θ
1
sin2 θ
2
= csc θ
Similarly, 1 + tan θ = sec θ can be obtained by rewriting the left side of this identity in terms of sine and
cosine. This gives
1 + tan θ = 1 +
Rewrite left side.
+
Write both terms with the common denominator.
=
(2)
=
2
2
2
=
sin θ 2
cos θ
sin θ 2
cos θ 2
cos θ
cos θ
cos2 θ+sin2 θ
cos2 θ
1
cos2 θ
2
= sec θ
The next set of fundamental identities is the set of even-odd identities. The even-odd identities relate
the value of a trigonometric function at a given angle to the value of the function at the opposite angle and
determine whether the identity is odd or even. (See Table 2).
Even-Odd Identities
tan (−θ) = −tan θ
sin (−θ) = −sin θ
cos (−θ) = cos θ
cot (−θ) = −cot θ
csc (−θ) = −csc θ
sec (−θ) = sec θ
Table 2
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Recall that an odd function is one in which f (−x) = −f (x) for all x in the domain of f. The sine
function is an odd function because sin (−θ) = −sin θ. The graph of an odd function is symmetric
about
the origin. For example,
consider
corresponding
inputs
of
and
−
.
The
output
of
sin
is
opposite
the
output of sin − . Thus,
π
2
π
2
π
2
=1
− π2
= −sin
sin
sin
Graph of
π
2
and
π
2
(3)
= −1
This is shown in Figure 2.
Figure 2:
π
2
y = sin θ
Recall that an even function is one in which
f (−x) = f (x) for all x in the domain of f
(4)
The graph of an even function is symmetric about the axis. The cosine function is an even function
because cos (−θ) = cos θ. For example,
consider
corresponding
inputs
and
−
.
The
output
of
cos
is
the same as the output of cos − . Thus,
y-
π
4
π
4
cos − π4 = cos
See Figure 3.
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π
4
≈ 0.707
π
4
π
4
(5)
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Figure 3:
Graph of
5
y = cos θ
For all θ in the domain of the sine and cosine functions, respectively, we can state the following:
• Since sin (−θ) = −sin θ, sine is an odd function.
• Since, cos (−θ) = cos θ, cosine is an even function.
The other even-odd identities follow from the even and odd nature of the sine and cosine functions. For
example, consider the tangent identity, tan (−θ) = −tan θ. We can interpret the tangent of a negative angle
as tan (−θ) =
=
= −tan θ. Tangent is therefore an odd function, which means that tan (−θ) =
−tan (θ) for all θ in the domain of the tangent function.
The cotangent identity, cot (−θ) = −cot θ, also follows from the sine and cosine identities. We can
interpret the cotangent of a negative angle as cot (−θ) =
=
= −cot θ. Cotangent is therefore an
odd function, which means that cot (−θ) = −cot (θ) for all θ in the domain of the cotangent function.
The cosecant function is the reciprocal of the sine function, which means that the cosecant of a negative
=
= −csc θ. The cosecant function is therefore odd.
angle will be interpreted as csc (−θ) =
Finally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is
interpreted as sec (−θ) =
=
= sec θ. The secant function is therefore even.
To sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions
are odd, verifying the even-odd identities.
The next set of fundamental identities is the set of reciprocal identities, which, as their name implies,
relate trigonometric functions that are reciprocals of each other. See Table 3.
sin(−θ)
cos(−θ)
−sin θ
cos θ
cos(−θ)
sin(−θ)
1
sin(−θ)
1
cos(−θ)
cos θ
−sin θ
1
−sin θ
1
cos θ
Reciprocal Identities
sin θ =
cos θ =
tan θ =
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1
csc θ
1
sec θ
1
cot θ
csc θ =
sec θ =
cot θ =
1
sin θ
1
cos θ
1
tan θ
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Table 3
The nal set of identities is the set of quotient identities, which dene relationships among certain
trigonometric functions and can be very helpful in verifying other identities. See Table 4.
Quotient Identities
tan θ =
sin θ
cos θ
cot θ =
cos θ
sin θ
Table 4
The reciprocal and quotient identities are derived from the denitions of the basic trigonometric functions.
The Pythagorean identities are based on the properties of a right triangle.
cos θ + sin θ = 1
(6)
1 + cot θ = csc θ
(7)
1 + tan θ = sec θ
(8)
The even-odd identities relate the value of a trigonometric function at a given angle to the value
of the function at the opposite angle.
tan (−θ) = −tan θ
(9)
cot (−θ) = −cot θ
(10)
sin (−θ) = −sin θ
(11)
csc (−θ) = −csc θ
(12)
cos (−θ) = cos θ
(13)
sec (−θ) = sec θ
(14)
The reciprocal identities dene reciprocals of the trigonometric functions.
A General Note:
2
2
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2
2
2
2
sin θ =
1
csc θ
(15)
cos θ =
1
sec θ
(16)
tan θ =
1
cot θ
(17)
csc θ =
1
sin θ
(18)
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sec θ =
1
cos θ
(19)
cot θ =
1
tan θ
(20)
tan θ =
sin θ
cos θ
(21)
cot θ =
cos θ
sin θ
(22)
The quotient identities dene the relationship among the trigonometric functions.
Example 1
Graphing the Equations of an Identity
Graph both sides of the identity cot θ =
cot θ and y =
.
Solution
See Figure 4.
1
tan θ
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1
tan θ .
In other words, on the graphing calculator, graph y =
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Figure 4
Analysis
We see only one graph because both expressions generate the same image. One is on top of the
other. This is a good way to prove any identity. If both expressions give the same graph, then they
must be identities.
How To:
Given a trigonometric identity, verify that it is true.
1.Work on one side of the equation. It is usually better to start with the more complex side, as
it is easier to simplify than to build.
2.Look for opportunities to factor expressions, square a binomial, or add fractions.
3.Noting which functions are in the nal expression, look for opportunities to use the identities
and make the proper substitutions.
4.If these steps do not yield the desired result, try converting all terms to sines and cosines.
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Example 2
Verifying a Trigonometric Identity
Verify tan θcos θ = sin θ.
Solution
We will start on the left side, as it is the more complicated side:
tan θcos θ =
sin θ
cos θ
cos
θ = sin θ )cos θ
)cos θ
(23)
= sin θ
Analysis
This identity was fairly simple to verify, as it only required writing tan θ in terms of sin θ and cos θ.
Try It:
Exercise 3
(Solution on p. 18.)
Verify the identity csc θcos θtan θ = 1.
Example 3
Verifying the Equivalency Using the Even-Odd Identities
Verify the following equivalency using the even-odd identities:
(24)
(1 + sin x) [1 + sin (−x)] = cos2 x
Solution
Working on the left side of the equation, we have
(1 + sin x) [1 + sin (−x)] = (1 + sin x) (1 − sin x)
2
= 1 − sin x
= cos2 x
Example 4
Verifying a Trigonometric Identity Involving sec2 θ
Verify the identity
= sin θ
Solution
As the left side is more complicated, let's begin there.
sec2 θ−1
sec2 θ
2
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Since sin(−x)= − sin x
Dierence of squares
cos x = 1 − sin x
2
2
(25)
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sec2 θ−1
sec2 θ
10
=
sec θ = tan θ + 1
(tan2 θ+1)−1
2
2
sec2 θ
2
θ
= tan
sec2 θ
2
1
sec2 θ
2
= tan θ
= tan2 θ cos θ
2 sin θ
cos2 θ
= cos
2θ
2 = sin 2θ
)cos2 θ
cos2 θ =
tan θ =
2
(26)
1
sec2 θ
sin2 θ
cos2 θ
)cos θ
2
= sin θ
There is more than one way to verify an identity. Here is another possibility. Again, we can start
with the left side.
sec2 θ−1
sec2 θ
=
sec2 θ
sec2 θ
−
1
sec2 θ
(27)
= 1 − cos2 θ
= sin2 θ
Analysis
In the rst method, we used the identity sec θ = tan θ + 1 and continued to simplify. In the second
method, we split the fraction, putting both terms in the numerator over the common denominator.
This problem illustrates that there are multiple ways we can verify an identity. Employing some
creativity can sometimes simplify a procedure. As long as the substitutions are correct, the answer
will be the same.
2
2
Try It:
Exercise 6
Show that
(Solution on p. 18.)
cot θ
csc θ
= cos θ.
Example 5
Creating and Verifying an Identity
Create an identity for the expression 2tan θsec θ by rewriting strictly in terms of sine.
Solution
There are a number of ways to begin, but here we will use the quotient and reciprocal identities
to rewrite the expression:
2 tan θsec θ = 2
=
Thus,
=
1
sin θ
cos θ
cos θ
2 sin θ
cos2 θ
2 sin θ
1−sin2 θ
Substitute 1 − sin θ for cos
2tan θsec θ =
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2
2 sin θ
1 − sin2 θ
(28)
2
θ
(29)
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Example 6
Verifying an Identity Using Algebra and Even/Odd Identities
Verify the identity:
(30)
sin2 (−θ) − cos2 (−θ)
= cos θ − sin θ
sin (−θ) − cos (−θ)
Solution
Let's start with the left side and simplify:
sin2 (−θ)−cos2 (−θ)
sin(−θ)−cos(−θ)
=
[sin(−θ)]2 −[cos(−θ)]2
sin(−θ)−cos(−θ)
θ)2 −(cos θ)2
= (−sin
−sin θ−cos θ
θ)2 −(cos θ)2
= (sin
−sin θ−cos θ
θ)(sin θ+cos θ)
= (sin θ−cos
−(sin θ+cos θ)
(sin θ−cos θ)()sin θ+cos θ )
=
and cos (−x) = cos x
Dierence of squares
(31)
sin (−x) = −sin x
−()sin θ+cos θ )
= cos θ − sin θ
Try It:
Exercise 9
Verify the identity
(Solution on p. 18.)
sin2 θ−1
tan θsin θ−tan θ
=
sin θ+1
tan θ .
Example 7
Verifying an Identity Involving
Cosines
and Cotangents
Verify the identity: 1 − cos x 1 + cot x = 1.
Solution
We will work on the left side of the equation.
2
1 − cos2 x
2
2
x
1 + cot2 x = 1 − cos2 x 1 + cos
2
sin x
sin2 x cos2 x 2
= 1 − cos x sin2 x + sin2 x
2 x+cos2 x = 1 − cos2 x sin sin
2x
1 2
= sin x sin2 x
=1
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Find the common denominator.
(32)
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2 Using Algebra to Simplify Trigonometric Expressions
We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in
simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas
of algebra, such as the dierence of squares formula, the perfect square formula, or substitution, will simplify
the work involved with trigonometric expressions and equations.
For example, the equation (sin x + 1) (sin x − 1) = 0 resembles the equation (x + 1) (x − 1) = 0, which
uses the factored form of the dierence of squares. Using algebra makes nding a solution straightforward
and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic
patterns in trigonometric expressions or equations.
Another example is the dierence of squares formula, a − b = (a − b) (a + b) , which is widely used in
many areas other than mathematics, such as engineering, architecture, and physics. We can also create
our own identities by continually expanding an expression and making the appropriate substitutions. Using
algebraic properties and formulas makes many trigonometric equations easier to understand and solve.
2
2
Example 8
Writing the Trigonometric Expression as an Algebraic Expression
Write the following trigonometric expression as an algebraic expression: 2cos θ + cos θ − 1.
Solution
Notice that the pattern displayed has the same form as a standard quadratic expression, ax
bx + c. Letting cos θ = x, we can rewrite the expression as follows:
2
2
+
2x2 + x − 1
This expression can be factored as (2x + 1) (x − 1) . If it were set equal to zero and we wanted to
solve the equation, we would use the zero factor property and solve each factor for x. At this point,
we would replace x with cos θ and solve for θ.
(33)
Example 9
Rewriting a Trigonometric Expression Using the Dierence of Squares
Rewrite the trigonometric expression: 4 cos θ − 1.
Solution
Notice that both the coecient and the trigonometric expression in the rst term are squared, and
the square of the number 1 is 1. This is the dierence of squares. Thus,
2
2
4 cos2 θ − 1 = (2 cos θ) − 1
= (2 cos θ − 1) (2 cos θ + 1)
Analysis
If this expression were written in the form of an equation set equal to zero, we could solve each
factor using the zero factor property. We could also use substitution like we did in the previous
problem and let cos θ = x, rewrite the expression as 4x − 1, and factor (2x − 1) (2x + 1) . Then
replace x with cos θ and solve for the angle.
2
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(34)
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Try It:
Exercise 13
Rewrite the trigonometric expression: 25 − 9 sin
(Solution on p. 18.)
2
θ.
Example 10
Simplify by Rewriting and Using Substitution
Simplify the expression by rewriting and using identities:
(35)
csc2 θ − cot2 θ
Solution
We can start with the Pythagorean identity.
(36)
1 + cot2 θ = csc2 θ
Now we can simplify by substituting 1 + cot θ for csc θ. We have
2
2
(37)
csc2 θ − cot2 θ = 1 + cot2 θ − cot2 θ
=1
Try It:
Exercise 15
(Solution on p. 18.)
=
Use algebraic techniques to verify the identity:
(Hint: Multiply the numerator and denominator on the left side by 1 − sin θ.
cos θ
1+sin θ
1−sin θ
cos θ .
Access these online resources for additional instruction and practice with the fundamental
trigonometric identities.
• Fundamental Trigonometric Identities
• Verifying Trigonometric Identities
Media:
1
2
1
2
http://openstaxcollege.org/l/funtrigiden
http://openstaxcollege.org/l/verifytrigiden
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3 Key Equations
Pythagorean identities
sin2 θ + cos2 θ = 1
1 + cot2 θ = csc2 θ
1 + tan2 θ = sec2 θ
tan (−θ) = −tan θ
cot (−θ) = −cot θ
Even-odd identities
sin (−θ) = −sin θ
csc (−θ) = −csc θ
cos (−θ) = cos θ
sec (−θ) = sec θ
1
csc θ
cos θ = sec1 θ
tan θ = cot1 θ
csc θ = sin1 θ
sec θ = cos1 θ
1
cot θ = tan
θ
sin θ =
Reciprocal identities
Quotient identities
tan θ =
cot θ =
sin θ
cos θ
cos θ
sin θ
Table 5
4 Key Concepts
•
•
•
•
•
•
•
There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates
how expressions can be rewritten to simplify a problem.
Graphing both sides of an identity will verify it. See Example 1.
Simplifying one side of the equation to equal the other side is another method for verifying an identity.
See Example 2 and Example 3.
The approach to verifying an identity depends on the nature of the identity. It is often useful to begin
on the more complex side of the equation. See Example 4.
We can create an identity by simplifying an expression and then verifying it. See Example 5.
Verifying an identity may involve algebra with the fundamental identities. See Example 6 and Example 7.
Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques
throughout this text, as they consist of the fundamental rules of mathematics. See Example 8, Example 9, and Example 10.
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5 Section Exercises
5.1 Verbal
Exercise 16
(Solution on p. 18.)
We know g (x) = cos x is an even function, and f (x) = sin x and h (x) = tan x are odd functions.
What about G (x) = cos x, F (x) = sin x, and H (x) = tan x? Are they even, odd, or neither?
Why?
Exercise 17
Examine the graph of f (x) = sec x on the interval [−π, π] . How can we tell whether the function
is even or odd by only observing the graph of f (x) = sec x?
Exercise 18
(Solution on p. 18.)
After examining the reciprocal identity for sec t, explain why the function is undened at certain
points.
Exercise 19
All of the Pythagorean identities are related. Describe how to manipulate the equations to get
from sin t + cos t = 1 to the other forms.
2
2
2
2
2
5.2 Algebraic
For the following exercises, use the fundamental identities to fully simplify the expression.
Exercise 20
(Solution on p. 18.)
sin x cos x sec x
Exercise 21
sin (−x) cos (−x) csc (−x)
Exercise 22
(Solution on p. 18.)
tan xsin x + sec xcos2 x
Exercise 23
csc x + cos xcot (−x)
Exercise 24
(Solution on p. 18.)
cot t+tan t
sec(−t)
Exercise 25
3 sin3 t csc t + cos2 t + 2 cos (−t) cos t
Exercise 26
(Solution on p. 18.)
−tan (−x) cot (−x)
Exercise 27
−sin(−x)cos x sec x csc x tan x
cot x
Exercise 28
1+tan2 θ
csc2 θ
Exercise 29
tan x
csc2 x
+
tan x
sec2 x
Exercise 30
1−cos2 x
tan2 x
(Solution on p. 18.)
+ sin2 θ +
1
sec2 θ
1+tan x 1+cot x
−
1
cos2 x
+ 2 sin2 x
(Solution on p. 18.)
For the following exercises, simplify the rst trigonometric expression by writing the simplied form in terms
of the second expression.
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Exercise 31
tan x+cot x
;
csc x
cos x
sec x+csc x
1+tan x ;
sin x
Exercise 32
(Solution on p. 18.)
Exercise 33
cos x
1+sin x
+ tan x; cos x
Exercise 34
1
sin xcos x
(Solution on p. 19.)
− cot x; cot x
Exercise 35
1
1−cos x
−
cos x
1+cos x ;
csc x
Exercise 36
(Solution on p. 19.)
(sec x + csc x) (sin x + cos x) − 2 − cot x; tan x
Exercise 37
1
csc x−sin x ;
sec x
Exercise 38
1−sin x
1+sin x
−
1+sin x
1−sin x
Exercise 39
and tan x
; sec x and tan x
(Solution on p. 19.)
tan x; sec x
Exercise 40
(Solution on p. 19.)
sec x; cot x
Exercise 41
sec x; sin x
Exercise 42
(Solution on p. 19.)
cot x; sin x
Exercise 43
cot x; csc x
For the following exercises, verify the identity.
Exercise 44
(Solution on p. 19.)
cos x − cos3 x = cos x sin2 x
Exercise 45
cos x (tan x − sec (−x)) = sin x − 1
Exercise 46
1+sin2 x
cos2 x
=
(Solution on p. 19.)
1
cos2 x
Exercise 47
+
sin2 x
cos2 x
= 1 + 2 tan2 x
2
(sin x + cos x) = 1 + 2 sin xcos x
Exercise 48
cos2 x − tan2 x = 2 − sin2 x − sec2 x
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(Solution on p. 19.)
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5.3 Extensions
For the following exercises, prove or disprove the identity.
Exercise 49
1
1+cos x
−
1
1−cos(−x)
= −2 cot x csc x
Exercise 50
(Solution on p. 19.)
csc2 x 1 + sin2 x = cot2 x
Exercise
51
sec2 (−x)−tan2 x
tan x
Exercise 52
tan x
sec x sin (−x)
2+2 tan x
2+2 cot x
− 2 sin2 x = cos 2x
(Solution on p. 19.)
= cos2 x
Exercise 53
sec(−x)
tan x+cot x
= −sin (−x)
Exercise 54
1+sin x
cos x
=
(Solution on p. 19.)
cos x
1+sin(−x)
For the following exercises, determine whether the identity is true or false. If false, nd an appropriate
equivalent expression.
Exercise 55
cos2 θ−sin2 θ
1−tan2 θ
= sin2 θ
Exercise 56
3 sin2 θ + 4 cos2 θ = 3 + cos2 θ
Exercise 57
sec θ+tan θ
cot θ+cos θ
= sec2 θ
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(Solution on p. 19.)
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Solutions to Exercises in this Module
Solution to Exercise (p. 9)
1
sin θ
sin θ cos θ cos θ
cos θ sin θ
sin θ cos θ
sin θcos θ
sin θcos θ
csc θcos θtan θ =
=
=
(38)
=1
Solution to Exercise (p. 10)
cot θ
csc θ
=
cos θ
sin θ
1
sin θ
=
cos θ
sin θ
·
(39)
sin θ
1
= cos θ
Solution to Exercise (p. 11)
sin2 θ−1
tan θsin θ−tan θ
=
(sin θ+1)(sin θ−1)
tan θ(sin θ−1)
sin θ+1
tan θ
=
Solution to Exercise (p. 13)
This is a dierence of squares formula: 25 − 9 sin
2
Solution to Exercise (p. 13)
cos θ
1+sin θ
1−sin θ
1−sin θ
θ = (5 − 3 sin θ) (5 + 3 sin θ) .
=
cos θ(1−sin θ)
1−sin2 θ
θ)
= cos θ(1−sin
cos2 θ
θ
= 1−sin
cos θ
(40)
Solution to Exercise (p. 15)
All three functions, F, G, and H, are even.
This is because F (−x) = sin (−x) sin (−x) = (−sin x) (−sin x) = sin x = F (x) , G (−x) = cos (−x) cos (−x) =
cos xcos x = cos x = G (x) and H (−x) = tan (−x) tan (−x) = (−tan x) (−tan x) = tan x = H (x) .
Solution to Exercise (p. 15)
When cos t = 0, then sec t = , which is undened.
2
2
2
1
0
Solution to Exercise (p. 15)
sin x
Solution to Exercise (p. 15)
sec x
Solution to Exercise (p. 15)
csc t
Solution to Exercise (p. 15)
−1
Solution to Exercise (p. 15)
sec2 x
Solution to Exercise (p. 15)
sin2 x + 1
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OpenStax-CNX module: m49393
19
Solution to Exercise (p. 16)
1
sin x
Solution to Exercise (p. 16)
1
cot x
Solution to Exercise (p. 16)
tan x
Solution to Exercise (p. 16)
−4sec xtan x
Solution
to Exercise (p. 16)
q
±
1
cot2 x
±
1−sin2 x
sin x
+1
Solution
to Exercise (p. 16)
√
Solution to Exercise (p. 16)
Answers will vary. Sample proof:
cos x − cos3 x = cos x 1 − cos2 x
= cos xsin2 x
Solution to Exercise (p. 16)
Answers will vary. Sample proof:
1+sin2 x
cos2 x
=
1
cos2 x
+
sin2 x
cos2 x
= sec2 x + tan2 x = tan2 x + 1 + tan2 x = 1 + 2tan2 x
Solution to Exercise (p. 16)
Answers will vary. Sample proof:
cos2 x − tan2 x = 1 − sin2 x − sec2 x − 1 = 1 − sin2 x − sec2 x + 1 = 2 − sin2 x − sec2 x
Solution to Exercise (p. 17)
False
False
Solution to Exercise (p. 17)
Proved with negative and Pythagorean identities
Solution to Exercise (p. 17)
True 3 sin θ + 4 cos θ = 3 sin θ + 3 cos θ + cos θ = 3
Solution to Exercise (p. 17)
2
2
2
2
2
sin2 θ + cos2 θ + cos2 θ = 3 + cos2 θ
Glossary
Denition 1: even-odd identities
set of equations involving trigonometric functions such that if f (−x) = −f (x) ,the identity is odd,
and if f (−x) = f (x) ,the identity is even
Denition 2: Pythagorean identities
set of equations involving trigonometric functions based on the right triangle properties
Denition 3: quotient identities
pair of identities based on the fact that tangent is the ratio of sine and cosine, and cotangent is the
ratio of cosine and sine
Denition 4: reciprocal identities
set of equations involving the reciprocals of basic trigonometric denitions
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