After watching this video lesson, you will understand what a trigonometric identity is. You will also know the different kinds of identities you will come across.

Trigonometric Identities

This video lesson is about trigonometric identities. These are the true statements about trigonometric functions. You can think of these as definitions, if you will. They explain trig functions by using simpler trig terms.

Just like there are many definitions in the English language, there are many identities in the trig world. A simple math identity is 4 = 3 + 1. In trigonometry, a simple identity can be tangent = sine/cosine. Notice that both statements are true. Both have also been written in simpler math terms.

Different Kinds

Do you remember how I said that there are all kinds of trigonometric identities, just like there are all kinds of definitions? Well, just like we can group our English words into categories, such as nouns, verbs, and adverbs, we can cluster our trigonometric identities into groups.

And just like some words in the English language are more popular, some of our trigonometric identities are more commonly used. The ones that are the most common have already been categorized into seven well-known groups. Do you want to see what they are? I’ll show you along with the identities themselves:

First we have the basic identities. These are your basic definitions of your six trig functions:

trig identities

The second group is called the Pythagorean identities. These are called Pythagorean because they make use of the Pythagorean theorem, which says that for a right triangle a^2 + b^2 = c^2, where c is the hypotenuse and a and b are the legs:

trig identities

The next group is called the angle sum and difference identities. These tell you how you can break down a trig function that involves the addition or subtraction of two angles:

trig identities

The fourth group is called the double-angle identities. These tell you how you can simplify a trig function where the angle is doubled:

trig identities

The fifth group is called the half-angle identities. Similar to the double-angle identities, these tell you how you can simplify a trig function where your angle has been halved:

trig identities

For the sixth group, we have what is called the sum to product identities. These identities show you how to convert the sum or difference of two trig functions into the product of two trig functions:

trig identities

The seventh and last group is called the product to sum identities. Similar to the previous group, these show how you can go from the product of two trig functions to the sum or difference of two trig functions:

trig identities

What you have seen so far is just a small sampling of all the trig identities out there. But, if you can memorize these, then you are well on your way to becoming a trig expert because these are the most commonly used identities in trigonometry.

Uses

How are these identities used? Well, you will use these in solving problems on tests. You will encounter problems that ask you to recall a specific identity. You will also be asked to simplify a trig problem by substituting in these identities. And you will be asked to prove various trig statements with the aid of these identities.

So, what you will need is to couple your knowledge of these trig identities with your algebra problem-solving skills to successfully answer these kinds of questions. Mathematicians also use these trig identities in their calculations in higher math to solve problems that involve integration.

Example

Do you want to see a sample problem? Okay. I’ll show you one.

‘What does sin (x) csc (x) equal to?’

Well, we can solve this problem by substituting in one of our basic identities. We have a sine function and a cosecant function. We can substitute 1/sin (x) for the csc (x). When we do that, we see that the sines cancel each other. What are we left with? We are left with just a 1. So our answer is simply and easily 1.

Lesson Summary

Let’s review what we’ve learned so far. Trigonometric identities are the true statements about trigonometric functions. If we liken them to definitions, we see that we have a lot of trigonometric identities just like there are lots of definitions. We also have categories or groups, just like definitions. We have seven groups of identities that are the most commonly used. They are:

  • The basic identities
  • The Pythagorean identities
  • The angle sum and difference identities
  • The double-angle identities
  • The half-angle identities
  • The sum to product identities
  • The product to sum identities

You will use these identities in helping you to simplify and prove trig problems.

Learning Outcomes

Following this lesson, you’ll be able to:

  • Define trigonometric identities
  • Describe the seven most common groups of trig identities and their uses
  • Solve trig problems using identities