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 Concept, you'll be able to calculate trig functions like the one above.

### Watch This

James Sousa: Sum and Difference Identities for Tangent

### Guidance

In this Concept, 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:

#### Example A

Find the exact value of .

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Solution:
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Use the difference formula for tangent, with

To verify this on the calculator, and .

#### Example B

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

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Solution:
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Use the difference formula for tangent, with

#### Example C

Find the exact value of .

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Solution:
**
Use the difference formula for tangent, with

### Vocabulary

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Tangent Sum Formula:
**
The
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tangent sum formula
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relates the tangent of a sum of two arguments to a set of tangent functions, each containing one argument.

**
Tangent Difference Formula:
**
The
**
tangent difference formula
**
relates the tangent of a difference of two arguments to a set of tangent functions, each containing one argument.

### Guided Practice

1. Find the exact value for

2. Simplify

3. Find the exact value for

**
Solutions:
**

1.

2.

3.

### Concept Problem Solution

The Concept Problem asks you to find:

You can use the tangent difference formula:

to help solve this. Substituting in known quantities:

### Practice

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