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Pre-Calculus 12A
Section 6.1, 6.3, 6.4
Reciprocal, Quotient, and Pythagorean Identities
In mathematics, it is important to understand the difference between an equation and an identity.
 An equation is only true for certain value(s) of the variable. For example, the equation x 2  4  0 is true
only for x  2 .
 An identity is an equation that is true for all values of the variable. For example: the equation
 x  2
2
 x 2  4x  4 is true for all values of x, so we can call it an identity.
A trigonometric identity is a trigonometric equation that is true for all permissible values of the variable.

An example of a trigonometric equation is sin x = 1, which is true only for x  90  360n,n  I

An example of a trigonometric identity is sin2 x  cos2 x  1 , which is true for all values of x.

An example of a trigonometric identity with non-permissible values is csc  
values of x except   0  180n,n  I , which is where sin  0 .
1
, which is true for all
sin 
You are already familiar with the reciprocal and quotient identities.
RECIPROCAL IDENTITIES
sin  
1
csc 
cos  
1
sec 
tan  
1
cot 
csc  
1
sin 
sec  
1
cos 
cot  
1
tan 
QUOTIENT IDENTITIES
tan  
sin 
cos 
cot  
cos 
sin 
These identities will be used to prove other trigonometric identities.
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Pre-Calculus 12A
Section 6.1, 6.3, 6.4
Example 1: Verify a Potential Identity Numerically
Consider sec  
tan 
sin 
a. Determine the non-permissible values, in degrees, for the identity sec  
b. Numerically verify that   60 and  
tan 
c. Prove that sec  
is an identity.
sin 

4
tan 
sin 
are solutions to the equation sec  
tan 
.
sin 
Solution:
a. To determine the non-permissible values, assess each trigonometric function in the equation individually
and examine expressions that may have non-permissible values.
Non-permissible values of the variable are values for which denominators are zero or numerators are
undefined.
sec  
Since sec  
undefined.
tan 
sin 
Left Hand Side
Right Hand Side
Consider sec 
tan
1
, cos   0 as division by 0 is
cos 
tan is not defined at 90  180n . Thus
90  180n are non-permissible values for the
variable.
What values of  would cause cos   0 ?
The non-permissible values of  would be
________________________.
sin
Since the denominator on the RHS is sin , you
must also consider what values of  would cause
sin  0 .
The non permissible values of  are;
______________________________________.
Combined, the non-permissible values for  are:
______________________________________________
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Pre-Calculus 12A
Section 6.1, 6.3, 6.4
b. Verify the equation sec  
60
sec 
sec 60
1
cos 60
1
0.5
2
sec  
tan 

is valid for x = 60 and
sin 
4
tan 
sin 
tan 
sin 
tan 60
sin 60
3
3
2
2
LHS=RHS
The equation is true for   60 .

4
sec 

4
1

cos
4
1
1
2
2
sec
sec  
tan 
sin 
tan 
sin 

4

sin
4
1
1
2
2
tan
LHS=RHS
The equation is true for  

4
.
Verifying that an equation is valid for a particular value of the variable is not sufficient to conclude that the
equation is an identity.
c. Prove that sec  
tan 
is an identity.
sin 
To prove that an identity is true for all permissible values, express both sides of the identity in equivalent forms.
One or both sides of the identity must be algebraically manipulated into an equivalent form to match the other
side.
There is a major difference between solving a trigonometric equation and proving a trigonometric identity:

Solving a trigonometric equation determines the value(s) that make that particular equation true. You
perform operations across the == sign to solve for the variable.

Proving a trigonometric identity shows that the expressions on each side of the =sign are equivalent.
You work on each side of the identity independently, and you do not perform operations across the
= sign.
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Pre-Calculus 12A
Section 6.1, 6.3, 6.4
sec  
tan 
sin 
LHS
sec 
RHS
tan 
sin 
Example 2: Use Identities to Simplify Expressions
a. Determine the non-permissible values, in radians, of the variable in the expression
b. Simplify the expression.
tan x cos x
sec x cot x
Solution:
a. The non-permissible values of tan x are___________________________________
The non-permissible values of sec x are _________________________________
The non-permissible values of cot x are __________________________________.
Combined, the non-permissible values for
tan x cos x
are ____________________
sec x cot x
b. To simplify the expression, use the reciprocal and quotient identities to rewrite the expression in terms of
cosine and sine.
Simplify
tan x cos x
sec x cot x
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Pre-Calculus 12A
Section 6.1, 6.3, 6.4
Example 3: Using the Reciprocal and Quotient Identities
Prove each of the following identities for all permissible values of x.
tan x cos x
cos2 x  sin x
a.
b. cot x  sec x 
1
sin x
sin x cos x
Solution:
a.
b.
LHS
tan x cos x
sin x
tan x cos x
1
sin x
cot x  sec x 
RHS
1
LHS
cot x  sec x
cos2 x  sin x
sin x cos x
RHS
2
cos x  sin x
sin x cos x
The Pythagorean Identity
Development of the Pythagorean Identity:
In the diagram to the left note the Pythagorean Theorem:
a2  b 2  c 2 .
Replace these values with x, y and r.
____________________________________
In the unit circle r = 1, cos   x and sin  y , substitute these
values for x, y and r.
THE PYTHAGOREAN
IDENTITY__________________________________________
cos2   sin2   1
cos2  
sin2  
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Pre-Calculus 12A
Section 6.1, 6.3, 6.4
cos2   sin2   1
cos2   sin2   1
Divide each term by cos2 
Divide each term by sin2 
Now isolate each term
individually to find the other
versions of the identity.
Now isolate each term
individually to find the other
versions of the identity.
THE PYTHAGOREAN IDENTITIES
Example 4: Using the Pythagorean Identities
Prove each of the following identities for all permissible values of x.
a. cos x  sin2 x cos x  cos3 x
b. sec x  cos x  tan x sin x
c. tan x sin x  sec x  cos x
sec x
d. sin x 
cot x  tan x
Solution:
a.
b.
cos x  sin2 x cos x  cos3 x
LHS
cos x  sin2 x cos x
RHS
cos3 x
sec x  cos x  tan x sin x
LHS
sec x
RHS
cos x  tan x sin x
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Pre-Calculus 12A
Section 6.1, 6.3, 6.4
c.
d.
tan x sin x  sec x  cos x
LHS
tan x sin x
RHS
sec x  cos x
sin x 
LHS
sin x
sec x
cot x  tan x
RHS
sec x
cot x  tan x
Tips for Proving Trigonometric Identities

It is easier to simplify a more complicated expression than to make a simple expression more
complicated, so start with the more complicated side of the identity.

Use known identities to make substitutions.

Rewrite the expressions using sine and cosine only.

If a quadratic is present; consider the Pythagorean identities.

Factor expressions to cancel out terms or create other identities.

Expand expressions to create identities.

If you have fractions, think about making common denominators.

Multiply the numerator and denominator by the conjugate of an expression.

Always keep the “target expression” in mind. Continually refer to it.
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Pre-Calculus 12A
Section 6.1, 6.3, 6.4
Example 5: Solving Equations Using the Quotient, Reciprocal, and Pythagorean Identities
Solve for  in general form and then in the specified domain. Use exact solutions when possible. Check for
non-permissible values.
a. cos2   cot  sin , 0    2
b. 2sin  7  3csc  ,  360    180
c.
sin2   2cos   2,  2    2
Solution:
a.
cos2   cot  sin , 0    2
b. 2sin  7  3csc  ,  360    180
c.
sin2   2cos   2,  2    2
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