Trigonometric Identities

Putting the Puzzle Pieces Together

Establishing Identities

Two functions f and g are said to be identically equal if f(x) = g(x) for every value of x for which both functions are defined. Such an equation is referred to as an identity.

Establishing an Identity Use the basic trig identities to

establish the identity csc θ tan θ = sec θ

1 sin 1sec

sin cos cos

Guidelines for Establishing Identity

1. Start with the side containing the more complicated expression

2. Rewrite sums or differences of quotients as a single quotient

3. Sometimes, rewriting one side in terms of sines and cosines only will help

4. Always keep your goal in mind. Keep looking at the other side of the equation as you work

Establishing an Identity Establish the

identity2 2

2 2

2 2

sin ( ) cos ( ) ( )

( sin ) (cos )

(sin ) (cos )

1

even odd properties

Guidelines for Establishing Identities

Be careful not to handle like an equation. You cannot establish an identity by such methods as adding the same expression to each side and obtaining a true statement.

Establishing an Identity Establish the

identity

2 2

tan cot sec csc 0

sin cos 1 10

cos sin cos sin

sin cos 10

cos sin cos sin1 1

0cos sin cos sin0 0

Establishing an Identity Establish the

identity1 sin 1 sin

4 tan sec1 sin 1 sin

Solution

2 2

2

2

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

1 sin (1 sin )

(1 2sin sin ) (1 2sin sin )

(1 sin )

4sin4 tan sec

cos4sin 1

4 tan seccos cos

Establishing an Identity Establish the

identity3

2cos sin sincot cos

sin

Solution

3

2

2 2

cos sin sin

sin sin sin

cot 1 sin

cot cos cot cos

Establishing an Identity Establish the identity

1 sin cos

cos 1 sin

Solution

2

2

1 sin 1 sin

cos 1 sin

1 sin

cos (1 sin )

cos

cos (1 sin )

cos

(1 sin )

Establishing Inverse Trig Identities

1

2Show that sin(tan )

1

Steps:

1. Let the inverse trig function be equal to

2. Look at what value you are trying to find

3. Use pythagorean identities to arrive at results

vv

v

Establishing Trig Identities

Many more examples on pgs. 478 – 479

On-line examples

Solution

1

2 2

2 2

2

tan tan

sin

sin 1tan cos sin cos

cos sec

sin tan 1 secsec

sec tan 1 11

Let v v

Trying to find

v

v

vv so

v

Establishing Inverse Trig Identities Show that

1 2cos sin 1v v

Solution

1

2 2

2 2

2 2

sin sin

cos

sin cos 1

cos 1 sin

cos 1 sin cos 1

Let v v

Trying to find

v

One Last Toughie Establish the

identity

(tan tan )(1 cot cot ) (cot cot )(1 tan tan ) 0

1 1 1 1(tan tan ) 1 (1 tan tan ) 0

tan tan tan tan

1 1 1 1tan tan tan tan 0

tan tan tan tan

0 0

FOIL