Section 5.1
Verifying Trigonometric Identities
Overview
• In Chapter 4, we developed several classes of
trigonometric identities:
1. Quotient
2. Reciprocal
3. Pythagorean
4. Even-Odd
• Identities are true for all values of x for which
the trig functions are defined.
The Fundamental Identities
Verification
• When we verify a trig identity, we show that
one side of the identity can be simplified so
that it is identical to the other side.
Rules, Guidelines, and Suggestions
1.
2.
3.
4.
Start with the side that appears to be more complicated.
Re-write trig functions in terms of sines and cosines.
Apply fundamental identities.
Use algebraic techniques such as factoring or combining
like terms.
5. Use arithmetic techniques such as finding a common
denominator, separating fractional terms, or multiplying
by a conjugate.
6. Do NOT move terms from one side to the other!
7. Frowned on by many: As a last resort, work on both sides
separately.
Example 1: Changing to Sines and Cosines to Verify an
Identity
•Verify the identity: csc x tan x  sec x.
csc x tan x  sec x
1 sin x
csc x tan x 
sin x cos x
1
sin x
csc x 
; tan x 
sin x
cos x
1 sin x

sin x cos x
Divide the numerator and the
denominator by the common factor.
1

cos x
Multiply the remaining factors in
the numerator and denominator.
 sec x
1
 sec x
cos x
The identity is verified.
Example 2: Using Factoring to Verify an Identity
sin x  sin x cos 2 x  sin 3 x.
•Verify the identity:
sin x  sin x cos 2 x  sin 3 x
sin x  sin x cos 2 x  sin x(1  cos 2 x) Factor sin x from
the two terms.
 sin x(sin 2 x)
sin 2 x  cos 2 x  1
 sin x
Multiply.
The identity is verified.
3
sin 2 x  1  cos 2 x
Example 3: Combining Fractional Expressions (with
common denominator) to Verify an Identity
•Verify the identity:
sin x
1  cos x

 2csc x.
1  cos x
sin x
sin x
1  cos x

 2csc x
1  cos x
sin x
The least common denominator is
sin x(1 + cos x)
sin x
1  cos x
sin x(sin x)
(1  cos x)(1  cos x)



1  cos x
sin x
sin x(1  cos x)
sin x(1  cos x)
sin 2 x
1  2cos x  cos 2 x


sin x(1  cos x)
sin x(1  cos x)
Use FOIL to multiply
(1 + cos x)(1 + cos x)
Example 3: (continued)
Verify the identity:
sin x
1  cos x

 2csc x.
1  cos x
sin x
sin 2 x  1  2cos x  cos 2 x

sin x(1  cos x)
Add the numerators. Put this
sum over the LCD.
sin 2 x  cos 2 x  1  2cos x
Regroup terms in the numerator.

sin x(1  cos x)
1  1  2cos x

sin 2 x  cos 2 x  1
sin x(1  cos x)
2  2cos x
Add constant terms in the numerator.

sin x(1  cos x)
Example 3: (continued)
•Verify the identity:

sin x
1  cos x

 2csc x.
1  cos x
sin x
2 (1  cos x)
sin x (1  cos x)
2

sin x
1
2
sin x
 2csc x
Factor and simplify.
Factor out the constant term.
1
csc x 
sin x
The identity is verified.
Example 4: Using a Pythagorean Identity to Verify an
Identity
sec2 t
 tan t  cot t
tan t
tan 2 t  1

tan t
tan 2 t
1


tan t tan t
tan t  cot t 
Example 5 : Separating a Single-Term quotient into Two
Terms to Verify an Identity
1  sin x
cos x
 sec x  tan x
1
sin x


cos x cos x
sec x  tan x 
Examples—Verify The Following
6) sin x sec x  tan x
7) sec x  sec x sin 2 x  cos x
tan  cot 
8)
 sin 
csc 
csc 2 t
9)
 csc t sec t
cot t
More Examples
sin t cos t
10)

1
csc t sec t
11)
cos x 1  sin x

 2sec x
1  sin x
cos x
sin 2 x  cos 2 x
12)
 sin x  cos x
sin x  cos x