This tutorial assumes you are familiar with the trigonometric functions and their derivation from the unit circle.

A tutorial on the trigonometric functions can be found here.

A tutorial on the trigonometric functions and the unit circle can be found here.

# What is an Identity?

An identity (in mathematics) is something that is true (more precisely: *a tautological relationship*).

For example, the following is an identity:

`1 + 1 = 2`

It says that `1 + 1`

is __exactly__ the same as `2`

.

A common (and important) trigonometric identity is:

`sin`^{2}θ + cos^{2}θ = 1

All this identity says is that what is on the *left hand side* is __identical__ to what is on the *right hand side*.

# Why Learn / Use Identities?

Identities (in any branch of mathematics) help us to:

- solve
- simplify
- or gain insight into

mathematical problems.

Identities are a lot like synonyms in language. Instead of using the word *red* to describe the colour of something we could use one of the following: *carmine, cherry, copper, crimson, ruby, ruddy, rust, scarlet*. In language, synonyms give spice to communication; in mathematics, it is used to look at things in a different way.

For example, if you were solving an trigonometric problem and ended up with the following expression:

`cos`^{2}θ × tanθ × secθ × cscθ

you might not think you had achieved much. However, by using basic trigonometric identities, you would see (or maybe have to work out) that the entire expression can be reduced to the value of `1`

.

# How Many Trigonometric Identities are There?

There are an infinite number of trigonometric identities. Only a few are useful, the remainder generally ending up as problems and exercises for students to solve.

# Reference Image

Basic trigonometric identities fall into four groups:

- Reciprocals
- Ratios
- Products
- Pythagorean Theorem

# Reciprocal Identities

These identities derive from the fundamental definitions of the trigonometric functions:

Given a right angle triangle, we know that:

`sinθ = A / C`

We also know that:

`cscθ = C / A`

It should be obvious that `sinθ = 1 / cscθ`

because the ratio `C / A`

(cosecant) is the reciprocal of the ratio `A / C`

(sine).

The same reasoning applies to all the __reciprocal__ identities.

# Ratio Identities

These identities also derive from the fundamental definitions of trigonometric functions and are very similar to the __reciprocal__ identities, except that they consist of the ratio of two trigonometric functions:

The most important of these is probably `tanθ = sinθ / cosθ`

(highlighted in blue).

We know that

```
tanθ = A / B (from a right angle triangle)
tanθ = y / x (from the unit circle)
```

We also know that `tanθ`

is the slope of the hypotenuse.

From the unit circle we know that:

```
sinθ = y
cosθ = x
```

It is correct to say that `sinθ = y`

is an __identity__ because both are equivalent or identical.

Using the identities `sinθ`

and `cosθ`

to substitute for `y`

and `x`

, we get the following:

`tanθ = y / x = sinθ / cosθ`

A similar derivation can be done using the definitions from a right angle triangle:

```
sinθ = A / C
cosθ = B / C
```

In order to be able to substitute our identities, we need to multiply both the numerator and denominator by `1 / C`

One of the principles of algebra is that as long as you multiply or divide ALL the terms by the same amount, the equation remains the same)

```
A (1 / C) (A / C) sinθ
tanθ = - × ——————— = ——————— = ————
B (1 / C) (B / C) cosθ
```

The same sort of reasoning applies to the derivation of the remaining __ratio__ identities.

Another class of *obvious* __ratio__ identities is the ratio of the term to itself:

```
sinθ
———— = 1
sinθ
```

In all cases, the answer is `1`

.

# Product Identities

While these can be derived from the fundamental trigonometric identities, it is often easier to use the __ratio__ identities.

Taking `cosθ × tanθ = sinθ`

, we can derive it as follows:

**1) Find** an identity for `tanθ`

:

```
sinθ
tanθ = ————
cosθ
```

**2) Substitute** this *identity* for `tanθ`

into the identity `cosθ × tanθ = sinθ`

:

```
sinθ
cosθ × ———— = sinθ
cosθ
```

**3) Cancel** the `cosθ`

terms:

```
sinθ
```~~cosθ~~ × ———— = sinθ
~~cosθ~~

The same sort of reasoning applies to the derivation of the remaining __product__ identities.

Another class of *obvious* __product__ identities is the product of the term to its __reciprocal__ identity:

`sinθ × cscθ = 1`

In all cases, the answer is `1`

.

# Pythagorean Theorem Identities

Because the trigonometric functions are derived from a right angle triangle and because the trigonometric functions make use of the sides of a right angle triangle, they can be used in the Pythagorean Theorem:

`A`^{2} + B^{2} = C^{2}

### sin^{2}θ + cos^{2}θ = 1

This is the most common (or most famous) example of a Pythagorean Theorem identity using trigonometric functions.

It can be arranged in many different ways, the three most common would be:

`sin`^{2}θ + cos^{2}θ = 1
sin^{2}θ = 1 - cos^{2}θ
cos^{2}θ = 1 - sin^{2}θ

Looking at a right angle triangle, we see that it has 3 sides, normally labelled A, B, and C. With C being the hypotenuse.

Using these sides in the Pythagorean Theorem, we get:

`A`^{2} + B^{2} = C^{2}

To get it to look like `sin`

, we see that we need to get ^{2}θ + cos^{2}θ = 1`C`

to be equal to ^{2}`1`

. This can be done by dividing the entire equation by `C`

:^{2}

`A`^{2} B^{2} C^{2}
— + — = —
C^{2} C^{2} C^{2}

Which gives us:

`A`^{2} B^{2}
— + — = 1
C^{2} C^{2}

This can be rewritten as:

```
A A B B
— × —
```**+** — × — = 1
C C C C

We know that `A/C = sinθ`

and `B/C = cosθ`

, so we can substitute them in the equation giving us:

`sinθ × sinθ + cosθ × cosθ = 1`

Which, when we collect the terms, gives us the familiar:

`sin`^{2}θ + cos^{2}θ = 1

### 1 + cot^{2}θ = csc^{2}θ

Just as we obtained `sin`

by dividing the terms in the Pythagorean Theorem by ^{2}θ + cos^{2}θ = 1`C`

, we can divide all the terms in the Pythagorean Theorem by ^{2}`A`

, which gives us:^{2}

`A`^{2} B^{2} C^{2}
— + — = —
A^{2} A^{2} A^{2}

By using similar steps as above, we get the identity:

`1 + cot`^{2}θ = csc^{2}θ

Three more common arrangements would be:

`cot`^{2}θ = csc^{2}θ - 1
csc^{2}θ = cot^{2}θ + 1
csc^{2}θ - cot^{2}θ = 1

### tan^{2}θ + 1 = sec^{2}θ

Just as we obtained `sin`

by dividing the terms in the Pythagorean Theorem by ^{2}θ + cos^{2}θ = 1`C`

, we can divide all the terms in the Pythagorean Theorem by ^{2}`B`

, which gives us:^{2}

`A`^{2} B^{2} C^{2}
— + — = —
B^{2} B^{2} B^{2}

By using similar steps as above, we get the identity:

`tan`^{2}θ + 1 = sec^{2}θ

Three common arrangements would be:

`tan`^{2}θ = sec^{2}θ - 1
sec^{2}θ = tan^{2}θ + 1
sec^{2}θ - tan^{2}θ = 1

# Trigonometric Identity Problems

All of the following are trigonometric identities. Feel free to verify / prove using the identities described in the tutorial:

Note, I have used different terms to indicate the angles. This is because the actual term used does not matter.

` csc`^{2}θ
————————— = cot^{2}θ
tan^{2}θ + 1

`(1 - sin`^{2}α)(1 + tan^{2}α) = 1

`sec`^{2}φ - 1
————————— = sin^{2}φ
sec^{2}φ

`sinβ + cosβcotβ = cscβ`

`1 - sin`^{4}γ - 2sin^{2}γcos^{2}γ = cos^{4}γ

It should be obvious that you can create as many identities as you like. It should also be obvious that most (all?) of these are not as useful as the fundamental identities described in this tutorial.

Is the following an identity?

`sin`^{2}θ + cos^{2}β = 1

It is an identity __if and only if__ θ = β, otherwise, it is not an identity.

## Leave a Reply