Suppose you were given two angles and asked to find the tangent of the difference of them. For example, can you compute:

Would you just subtract the angles and then take the tangent of the result? Or is something more complicated required to solve this problem? Keep reading, and by the end of this lesson, you'll be able to calculate trig functions like the one above.

### Tangent Sum and Difference Formulas

In this lesson, we want to find a formula that will make computing the tangent of a sum of arguments or a difference of arguments easier. As first, it may seem that you should just add (or subtract) the arguments and take the tangent of the result. However, it's not quite that easy.

To find the sum formula for tangent:

In conclusion, . Substituting for in the above results in the difference formula for tangent:

#### Using the Tangent Difference Formula

1. Find the exact value of .

Use the difference formula for tangent, with

To verify this on the calculator, and .

2. Verify the tangent difference formula by finding , since this should be equal to .

Use the difference formula for tangent, with

3. Find the exact value of .

Use the difference formula for tangent, with

### Examples

#### Example 1

Earlier, you were asked to find .

You can use the tangent difference formula:

to help solve this. Substituting in known quantities:

#### Example 2

Find the exact value for

#### Example 3

Simplify

#### Example 4

Find the exact value for

### Review

Find the exact value for each tangent expression.

Write each expression as the tangent of an angle.

- Prove that
- Prove that
- Prove that
- Prove that

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.8.