Section 5.1

Verifying Trigonometric Identities

Overview

• In Chapter 4, we developed several classes of trigonometric identities:

1.Quotient2.Reciprocal3.Pythagorean4.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 Suggestions1. Start with the side that appears to be more complicated.2. Re-write trig functions in terms of sines and cosines.3. Apply fundamental identities.4. 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 tan sec .x x x

csc tan secx x x

1 sinsi

cn cos

sc tan xx

xx

x

Divide the numerator and thedenominator by the common factor.

1

sin x sin x

cos x

1cos x

sec x

1 sincsc ; tan

sin cosx

x xx x

Multiply the remaining factors inthe numerator and denominator.

The identity is verified.1

seccos

xx

Example 2: Using Factoring to Verify an Identity

•Verify the identity: 2 3sin sin cos sin . x x x x

2 3sin sin co s ns i x x x x22sin sin co sin (1s cos ) xx x xx

2sin (sin )x x

3sin x

Factor sin x fromthe two terms.

2 2

2 2

sin cos 1

sin 1 cos

x x

x x

Multiply.The identity is verified.

Example 3: Combining Fractional Expressions (with common denominator) to Verify an Identity

•Verify the identity: sin 1 cos1 cos sin

2csc .

x xx x

x

sin 1 co2c

s1 cos i

sn

cs

x x

x xx

sin 1 cos sin (sin ) (1 cos )(1 cos )1 cos sin sin (1 cos ) sin (1 cos )

x x x x x xx x x x x x

2 2sin 1 2cos cossin (1 cos ) sin (1 cos )

x x xx x x x

The least common denominator issin x(1 + cos x)

Use FOIL to multiply(1 + cos x)(1 + cos x)

Example 3: (continued)

2 2sin 1 2cos cossin (1 cos )x x x

x x

2 2sin cos 1 2cos

sin (1 cos )x x x

x x

1 1 2cos

sin (1 cos )x

x x

2 2cos

sin (1 cos )x

x x

Add the numerators. Put this sum over the LCD.

Regroup terms in the numerator.

2 2sin cos 1x x

Add constant terms in the numerator.

Verify the identity: sin 1 cos2csc .

1 cos sin

x x

xx x

Example 3: (continued)

•Verify the identity:

2 (1 cos )x

sin (1 cos )x x2

sin x

12

sin x

2csc x

sin 1 cos2csc .

1 cos sin

x x

xx x

Factor and simplify.

Factor out the constant term.

1csc

sinx

x The identity is verified.

Example 4: Using a Pythagorean Identity to Verify an Identity

2sec

tac

ntan ot t

t

tt

t

t

tan

1tan2

tt

t

tan

1

tan

tan2

tt cottan

Example 5 : Separating a Single-Term quotient into Two Terms to Verify an Identity

sec tan

1 sin

cos cos

se

1 sin

cos

c tan

x x

x

x

x

x

x

x

x

Examples—Verify The Following

2

2

6)sin sec tan

7)sec sec sin cos

tan cot8) sin

csc

csc9) csc sec

cot

x x x

x x x x

tt t

t

More Examples

2 2

sin cos10) 1

csc sec

cos 1 sin11) 2sec

1 sin cos

sin cos12) sin cos

sin cos

t t

t t

x xx

x x

x xx x

x x