An identity is an equation that is always true no matter what values are chosen. We use the sign ≡ to indicate that we have an identity. An example of an identity is shown below:

You may be asked in the exam to prove that an identity holds. We do this by manipulating both sides of the equation so that they are the same. Let’s have a few examples.

**Example 1**

Prove the identity below:

Both sides of the above equations have brackets. Therefore, the best way to prove the above identity is to multiply out all of the brackets for both sides, and then simplify the terms on both sides of the equations.

We multiply out the brackets by multiplying the number that is on the outside of the bracket by all of the terms that are on the inside of the bracket. We need to be careful when multiplying the second bracket on the left and this is because the number on the outside of the bracket is -3 and we need to make sure that we remember the negative sign. The multiplying out of both sides of the identity are shown below.

We multiply out the brackets by multiplying the number that is on the outside of the bracket by all of the terms that are on the inside of the bracket. We need to be careful when multiplying the second bracket on the left and this is because the number on the outside of the bracket is -3 and we need to make sure that we remember the negative sign. The multiplying out of both sides of the identity are shown below.

The next step is to collect like terms; we collect the

*x*’s and we collect the numbers.We can now see that both sides of the identity are the same. This means that the identity holds.

**Example 2**

Prove the identity below.

The above identity is more complex than the first identity. The end result of proving the identity is to get both sides of the identity the same. Both sides of the identity have one fraction with a denominator of 4. Therefore, it is highly likely that both sides of the identity will be a fraction with a denominator of 4.

It is easier to answer these types of question by working with each of the sides separately. I am going to work with the left side of the equation first and I am going to start with multiplying out the double bracket. The easiest way to multiply out double brackets is to use FOIL, which stands for First, Outside, Inside and Last.

**Left Hand Side**It is easier to answer these types of question by working with each of the sides separately. I am going to work with the left side of the equation first and I am going to start with multiplying out the double bracket. The easiest way to multiply out double brackets is to use FOIL, which stands for First, Outside, Inside and Last.

We can simplify the numerator because there are two terms that contains

*x*’s.There are two different parts of the left side of the equation; the 2 and the fraction. I am going to combine both of these terms. We can view the 2 as currently having a denominator of 1.

We are adding fractions and when we add fractions, we need to make sure that the denominators of the fractions are the same. I am going to make both of the denominators 4. Currently, the denominator of the first fraction is 1, and in order to make it 4, we need to multiply by 4. As we are creating equivalent fractions, we need to multiply both the numerator and the denominator by 4. The working is shown below.

We can now carry out the calculation and simplify the numerator.

**Right Hand Side**

I am now going to work with the right side of the equation and I am going to start by multiplying out the double brackets on the numerator; I will multiply out the double brackets by using FOIL again (First, Outside, Inside, Last).

We can simplify the numerator by combining the numbers.

The next step is to add the fractions together. When we are adding fractions, we need to make sure that the denominators are the same. Currently, the denominators are 4 and 2. Both of these numbers go into 4, which means that we can make both of the denominators 4. The denominator of the first fraction is already 4, but the denominator of the second fraction is 2. We are able to make the denominator of the second fraction 4 by multiplying both the numerator and the denominator by 2. The working is shown below.

We are now able to add these fractions together.

The final step is to simplify the numerator by combing the two numbers.

I am now going to place the manipulated left and right side of the equation together.

Both sides of the equation are exactly the same, which means that this identity holds.