# 3.3.2: Simplifying Trigonometric Expressions using Sum and Difference Formulas


Simplify sine, cosine, and tangent of angles that are added or subtracted.

As Agent Trigonometry you are given this clue: $$\sin\left(\dfrac{\pi}{2}−x\right)$$. How could you simplify this expression to make solving your case easier?

### Simplifying Trigonometric Expressions

We can also use the sum and difference formulas to simplify trigonometric expressions.

The $$\sin a=−\dfrac{3}{5}$$ and $$\cos b=\dfrac{12}{13}$$. a is in the $$3^{rd}$$ quadrant and b is in the $$1^{st}$$. Let's find $$\sin(a+b)$$.

First, we need to find $$\cos a$$ and $$\sin b$$. Using the Pythagorean Theorem, missing lengths are 4 and 5, respectively. So, $$\cos a=−\dfrac{4}{5}$$ because it is in the 3rd quadrant and $$\sin b=\dfrac{5}{13}$$. Now, use the appropriate formulas.

\begin{aligned}\sin(a+b)& =\sin a \cos b+\cos a \sin b \\&=−\dfrac{3}{5}\cdot \dfrac{12}{13}+−\dfrac{4}{5} \cdot \dfrac{5}{13}=−\dfrac{56}{65} \end{aligned}

Now, using the information from the previous problem above, let's find $$\tan(a+b)$$.

From the cosine and sine of $$a$$ and $$b$$, we know that $$\tan a=\dfrac{3}{4}$$ and $$\tan b=\dfrac{5}{12}$$.

\begin{aligned} \tan(a+b)&=\dfrac{\tan a+\tan b}{1−\tan a\tan b} \\&=\dfrac{\dfrac{3}{4}+\dfrac{5}{12}}{1−\dfrac{3}{4}\cdot \dfrac{5}{12}} \\&=\dfrac{\dfrac{14}{12}}{\dfrac{11}{16}}\\&=\dfrac{56}{33}\end{aligned}

Finally, let's simplify $$\cos(\pi −x)$$.

Expand this using the difference formula and then simplify.

\begin{aligned} \cos(\pi −x)&=\cos\pi \cos x+\sin \pi \sin x\\ &=−1\cdot \cos x+0\cdot \sin x \\ &=−\cos x \end{aligned}

##### Example $$\PageIndex{1}$$

Earlier, you were asked to simplify $$\sin\left(\dfrac{\pi}{2}−x \right)$$.

Solution

You can expand the expression using the difference formula and then simplify.

\begin{aligned} \sin\left(\dfrac{\pi}{2}−x \right)&=\sin\dfrac{\pi}{2} \cos x−\cos\dfrac{\pi}{2} \sin x \\ &=1 \cdot cosx−0\cdot \sin x \\&=\cos x \end{aligned}

##### Example $$\PageIndex{2}$$

Using the information from the first problem above (where we found $$\sin(a+b)$$), find $$\cos(a−b)$$.

Solution

\begin{aligned} \cos(a−b)&=\cos a \cos b+\sin a \sin b=−\dfrac{4}{5} \cdot \dfrac{12}{13}+−\dfrac{3}{5} \cdot \dfrac{5}{13} \\ &=−\dfrac{63}{65} \end{aligned}

##### Example $$\PageIndex{3}$$

Simplify $$\tan(x+\pi)$$.

Solution

\begin{aligned} \tan(x+\pi )&=\dfrac{\tan x+\tan\pi}{−\tan x \tan\pi} \\ &=\dfrac{\tan x+0}{1−\tan 0} \\&=\tan x\end{aligned}

## Review

$$\sin a=−\dfrac{8}{17}$$, $$\pi \leq a<\dfrac{3 \pi}{2}$$ and $$\sin b=−\dfrac{1}{2}, \; \dfrac{3 \pi}{2}\leq b<2\pi$$. Find the exact trig values of:

1. $$\sin(a+b)$$
2. $$\cos(a+b)$$
3. $$\sin(a−b)$$
4. $$\tan(a+b)$$
5. $$\cos(a−b)$$
6. $$\tan(a−b)$$

Simplify the following expressions.

1. $$\sin(2\pi −x)$$
2. $$\sin\left(\dfrac{\pi}{2}+x\right)$$
3. $$\cos(x+\pi )$$
4. $$\cos\left(\dfrac{3 \pi}{2}−x\right)$$
5. $$\tan(x+2\pi)$$
6. $$\tan(x−\pi )$$
7. $$\sin\left(\pi 6−x\right)$$
8. $$\tan\left(\dfrac{\pi }{4}+x\right)$$
9. $$\cos\left(x−\dfrac{\pi }{3}\right)$$

Determine if the following trig statements are true or false.

1. $$\sin(\pi −x)=\sin(x−\pi )$$
2. $$\cos(\pi −x)=\cos(x−\pi )$$
3. $$\tan(\pi −x)=\tan(x−\pi )$$

## Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.13.

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