An identity is an equation that is true for all defined values of a variable.

We are going to use the identities that we have already established and establish others to "prove" or verify other identities. Let's summarize the basic identities we have.

RECIPROCAL IDENTITIES

1csc

sinx

x

1sec

cosx

x

1cot

tanx

x

QUOTIENT IDENTITIESsin

tancos

xx

x

coscot

sin

xx

x

1sin

cscx

x

1cos

secx

x

1tan

cotx

x

2 2sin cos 1x x Let’s look at the Fundamental Identity derived on page 445

Now to find the two more identities from this famous and oft used one.

2 2sin cos 1x x Divide all terms by cos2x

cos2x cos2x cos2xWhat trig function is this squared? 1 What trig function

is this squared?

2 2tan 1 secx x 2 2sin cos 1x x Divide all terms by sin2x

sin2x sin2x sin2x

What trig function is this squared?

1 What trig function is this squared?

2 21 cot cscx x

These three are sometimes called the Pythagorean Identities since the derivation of the fundamental theorem used the Pythagorean Theorem

All of the identities we learned are found on the back page of your book.

You'll need to have these memorized or be able to derive them for this course.

QUOTIENT IDENTITIESsin

tancos

xx

x

coscot

sin

xx

x

2 2tan 1 secx x

2 21 cot cscx x PYTHAGOREAN IDENTITIES

2 2sin cos 1x x

RECIPROCAL IDENTITIES

1csc

sinx

x

1sec

cosx

x

1cot

tanx

x

1sin

cscx

x

1cos

secx

x

1tan

cotx

x

One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this:

sintan

cos

xx

x

1sec

cosx

x

1csc

sinx

x

tan cscSimplify:

sec

x x

x

sin 1cos sin

1cos

xx x

x

substitute using each identity

simplify1

cos1

cos

x

x

1

Another way to use identities is to write one function in terms of another function. Let’s see an example of this:

2

Write the following expression

in terms of only one trig function:

cos sin 1x x This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.

2 2sin cos 1x x 2 2cos 1 sinx x

2= 1 sin sin 1x x

2= sin sin 2x x

A third way to use identities is to find function values. Let’s see an example of this:

2

Write the following expression

in terms of only one trig function:

cos sin 1x x This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.

2 2sin cos 1x x 2 2cos 1 sinx x

2= 1 sin sin 1x x

2= sin sin 2x x

1Given sin with in quadrant II,

3find the other five trig functions using identities.

We'd get csc by taking reciprocal of sin

csc 3Now use the fundamental trig identity1cossin 22

Sub in the value of sine that you know

1cos3

1 22

Solve this for cos

9

8cos2

8 2 2cos

39

When we square root, we need but determine that we’d need the negative since we have an angle in Quad II where cosine values are negative.

square root both sides

A third way to use identities is to find function values. Let’s see an example of this: 1

cscsin

2 2cos

3

3

1sin

csc 3

We need to get tangent using fundamental identities.

cos

sintan

Simplify by inverting and multiplying13tan

2 23

Finally you can find cotangent by taking the reciprocal of this answer.

3sec

2 2

1 3

3 2 2

1

2 2

cot 2 2

You can easily find sec by taking reciprocal of cos.

This can be rationalized

2

2 3 2

4

2

4

This can be rationalized

Now let’s look at the unit circle to compare trig functions of positive vs. negative angles.

?3

cos isWhat

?3

cos isWhat

Remember a negative angle means to go clockwise

2

1

2

1

2

3,

2

1

cos cosx x Recall from College Algebra that if we put a negative in the function and get the original back it is an even function.

?3

sin isWhat

?3

sin isWhat

2

3

2

3

2

3,

2

1

sin sinx x Recall from College Algebra that if we put a negative in the function and get the negative of the function back it is an odd function.

?3

tanisWhat

?3

tanisWhat

2

3,

2

1

3

3

If a function is even, its reciprocal function will be also. If a function is odd its reciprocal will be also.

EVEN-ODD PROPERTIESsin(- x ) = - sin x (odd) csc(- x ) = - csc x (odd)

cos(- x) = cos x (even) sec(- x ) = sec x (even)

tan(- x) = - tan x (odd) cot(- x ) = - cot x (odd)

angle? positivea of termsin what 60sin

60sin

angle? positivea of termsin what 3

2sec

3

2sec

RECIPROCAL IDENTITIES1

cscsin

xx

1

seccos

xx

1cot

tanx

x

QUOTIENT IDENTITIESsin

tancos

xx

x

coscot

sin

xx

x

2 2tan 1 secx x 2 21 cot cscx x

PYTHAGOREAN IDENTITIES2 2sin cos 1x x

EVEN-ODD IDENTITIES

sin sin cos cos tan tan

csc csc sec sec cot cot

x x x x x x

x x x x x x

COFUNCION IDENTITIES

cos)

2sin(

sin)2

cos(

cot)

2tan(

tan)2

cot(

csc)

2sec(

sec)2

csc(