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# Tangent Sum and Difference Formulas

## Tangent of a sum or difference related to a set of tangent functions.

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Tangent Sum and Difference Formulas

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

tan(12040)\begin{align*}\tan (120^\circ - 40^\circ)\end{align*}

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:

tan(a+b)tan(a+b)=sin(a+b)cos(a+b)=sinacosb+sinbcosacosacosbsinasinb=sinacosb+sinbcosacosacosbcosacosbsinasinbcosacosb=sinacosbcosacosb+sinbcosacosacosbcosacosbcosacosbsinasinbcosacosb=sinacosa+sinbcosb1sinasinbcosacosb=tana+tanb1tanatanbUsing tanθ=sinθcosθSubstituting the sum formulas for sine and cosineDivide both the numerator and the denominator by cosacosbReduce each of the fractionsSubstitute sinθcosθ=tanθSum formula for tangent

In conclusion, tan(a+b)=tana+tanb1tanatanb\begin{align*}\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\end{align*}. Substituting b\begin{align*}-b\end{align*} for b\begin{align*}b\end{align*} in the above results in the difference formula for tangent:

tan(ab)=tanatanb1+tanatanb

#### Example A

Find the exact value of tan285\begin{align*}\tan 285^\circ\end{align*}.

Solution: Use the difference formula for tangent, with 285=33045\begin{align*}285^\circ = 330^\circ - 45^\circ\end{align*}

tan(33045)=tan330tan451+tan330tan45=3311331=3333=33333+33+3=963393=12636=23

To verify this on the calculator, \begin{align*}\tan 285^\circ = -3.732\end{align*} and \begin{align*}-2 -\sqrt{3} = -3.732\end{align*}.

#### Example B

Verify the tangent difference formula by finding \begin{align*}\tan \frac{6\pi}{6}\end{align*}, since this should be equal to \begin{align*}\tan \pi = 0\end{align*}.

Solution: Use the difference formula for tangent, with \begin{align*}\tan \frac{6\pi}{6} = \tan (\frac{7 \pi}{6} - \frac{\pi}{6})\end{align*}

#### Example C

Find the exact value of \begin{align*}\tan 165^\circ\end{align*}.

Solution: Use the difference formula for tangent, with \begin{align*}165^\circ = 225^\circ - 60^\circ\end{align*}

### Guided Practice

1. Find the exact value for \begin{align*}\tan 75^\circ\end{align*}

2. Simplify \begin{align*}\tan(\pi + \theta)\end{align*}

3. Find the exact value for \begin{align*}\tan 15^\circ\end{align*}

Solutions:

1.

2. \begin{align*}\tan (\pi + \theta) = \frac{\tan \pi + \tan \theta}{1- \tan \pi \tan \theta} = \frac{\tan \theta}{1} = \tan \theta\end{align*}

3.

### Concept Problem Solution

The Concept Problem asks you to find:

\begin{align*}\tan (120^\circ - 40^\circ)\end{align*}

You can use the tangent difference formula:

to help solve this. Substituting in known quantities:

### Explore More

Find the exact value for each tangent expression.

1. \begin{align*}\tan\frac{5\pi}{12}\end{align*}
2. \begin{align*}\tan\frac{11\pi}{12}\end{align*}
3. \begin{align*}\tan-165^\circ\end{align*}
4. \begin{align*}\tan255^\circ\end{align*}
5. \begin{align*}\tan-15^\circ\end{align*}

Write each expression as the tangent of an angle.

1. \begin{align*}\frac{\tan15^\circ+\tan42^\circ}{1-\tan15^\circ\tan42^\circ}\end{align*}
2. \begin{align*}\frac{\tan65^\circ-\tan12^\circ}{1+\tan65^\circ\tan12^\circ}\end{align*}
3. \begin{align*}\frac{\tan10^\circ+\tan50^\circ}{1-\tan10^\circ\tan50^\circ}\end{align*}
4. \begin{align*}\frac{\tan2y+\tan4}{1-\tan2\tan4y}\end{align*}
5. \begin{align*}\frac{\tan x-\tan3x}{1+\tan x\tan3x}\end{align*}
6. \begin{align*}\frac{\tan2x-\tan y}{1+\tan2x\tan y}\end{align*}
7. Prove that \begin{align*}\tan(x+\frac{\pi}{4})=\frac{1+\tan(x)}{1-\tan(x)}\end{align*}
8. Prove that \begin{align*}\tan(x-\frac{\pi}{2})=-\cot(x)\end{align*}
9. Prove that \begin{align*}\tan(\frac{\pi}{2}-x)=\cot(x)\end{align*}
10. Prove that \begin{align*}\tan(x+y)\tan(x-y)=\frac{\tan^2(x)-\tan^2(y)}{1-\tan^2(x)\tan^2(y)}\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 3.8.

### Vocabulary Language: English

Tangent Difference Formula

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.
Tangent Sum Formula

Tangent Sum Formula

The tangent sum formula relates the tangent of a sum of two arguments to a set of tangent functions, each containing one argument.