Section 4.4 – Trigonometric Expressions and Identities 1

Section 4.4

Trigonometric Expressions and Identities

In this section we are going to practice the algebra involved in working with the trigonometric

functions. This will help to pave the way for the more analytical parts of trigonometry in the

next chapter.

An identity is an equation that is satisfied by all relevant values of the variables concerned.

An example of an identity is 22))(( yxyxyx .

In simplifying identities it may be useful to use the following.

Basic Trigonometric Identities

1. 1cossin 22

2.

tan

cos

sin

3.

cos

1sec ;

sin

1csc ;

tan

1cot (Reciprocal Identities)

4. Pythagorean Identities:

1)(cos)(sin 22

)(sec1)(tan 22

)(csc1)(cot 22

Example 1: If =

and

, find the value of and .

Notational Conventions

1. An expression such as sin really means sin( ), where sin or sine is the name of the function

and is an input. An exception to this, however, occurs in expressions such as sin(A + B),

where the parentheses are necessary.

2. Parentheses are often omitted in multiplication. For example, the product (sin )(cos ) is

usually written sin cos .

3. The quantity (sin ) n is usually written sin n .

Section 4.4 – Trigonometric Expressions and Identities 2

Example 2: Multiply. 6)sec(5)sec(

Example 3: Factor. .12cot7cot 2

Example 4: Simplify.

a. )csc()tan(

b. )cot)(csccos1( xxx

c. 1csc

sin12

2

Section 4.4 – Trigonometric Expressions and Identities 3

d. )(sec

)cot()tan(2

e. x

x

x

x

sin1

cos

sin1

cos

f.

Section 4.4 – Trigonometric Expressions and Identities 4

Proving Identities

Some helpful hints when proving identities:

1. Your goal is to make one side look like the other side. You may want to start with the “most

complicated looking” side.

2. Get common denominators.

3. Convert everything to sine and cosine (this frequently works, but sometimes makes things

worse).

4. If you get stuck working from one side, try starting with the other.

Example 4: Prove the following identities.

a. 1costancsc AAA

b. sintancossec

Section 4.4 – Trigonometric Expressions and Identities 5

c. 2sin)cos(seccos

d. x

xxx

2

222

tan1

tan1sincos

Section 4.4 – Trigonometric Expressions and Identities 6

Example 5: Simplify

a.

b.