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Pre-Calculus 12A
Section 6.1
Reciprocal, Quotient, and Pythagorean Identities
It is critical to have a clear understanding of the difference between equations and identities. Equations can be
solved for certain value(s) of the variable. For example, the equation 3x = 12 is only true when x = 4.
Identity: an identity is an equation that is true for all values of the variables. For example:
x  y 
2
 x 2  2xy  y 2
This equation is true for all possible values of x and y, so it is called an identity.
Trigonometric identity: a trigonometric equation that is true for all permissible values of the variable in the
expressions on both sides of the equation.
Non-permissible values of trigonometric identities occur when there is a denominator in the identity. The nonpermissible values of any trigonometric identity can be determined by finding all values of  that would cause a
denominator of zero.
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, is not defined at sin  0 which occurs at 0 and 180
sin 
and every subsequent rotation of 180 . Therefore the non-permissible values for this identity in degrees would
be   0  180n, n  I . In radians   0   n,n  I .
For example, the trigonometric identity, csc  
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 
Example 1: Verify a Potential Identity Numerically
a. Determine the non-permissible values, in degrees, for the equation sec  
b. Numerically verify that   60 and  
c. Verify the identity algebraically.

4
tan 
sin 
are solutions of the equation.
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Pre-Calculus 12A
Section 6.1
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.
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 . So the nonpermissible vales of  are __________________
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;
______________________________________.
The three sets of non permissible values can be expressed as
______________________________________________
b. Verify numerically.
tan 
sec  
60
sin 
sec 
tan 
sin 
sec 60
tan 60
1
sin 60
cos 60
3
1
0.5
3
2
2
2
LHS=RHS

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
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Pre-Calculus 12A
Section 6.1
c. Verify algebraically.
sec 
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 identity for sec x and the quotient identities for tan x and
cot x to write the trigonometric expressions 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
Example 3: Using the Reciprocal and Quotient Identities
Prove each of the following identities:
tan x cos x
a.
1
sin x
b. cot x  sec x 
cos2 x  sin x
sin x cos x
Solution:
tan x cos x
sin x
1
cot x  sec x
cos2 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
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:
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:
cos x  sin2 x cos x
cos3 x
sec x
cos x  tan x sin x
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Pre-Calculus 12A
tan x sin x
Section 6.1
sec x  cos x
sin x
sec x
cot x  tan x
Strategies That May Assist in Proving Trigonometric Identities
1. Start by working on whatever side looks more complicated. Then work toward the less complicated
expression.
2. Rewrite everything in terms of sine and cosine.
3. If you have fractions, think about making common denominators.
4. Factor expressions to cancel out terms or create other identities.
5. Expand expressions to create identities.
6. When there is a square in the term, look for a way to apply the Pythagorean Identity.
7. Always keep the “target expression” in mind. Continually refer to it.
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