# Trigonometric Identities and Equations

## Sum and Difference Identities

How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long distances.
The trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are special equations or postulates, true for all values input to the equations, and with innumerable applications.
In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the term formula is used synonymously with the word identity.

### Using the Sum and Difference Formulas for Cosine

Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles, which we can review in the unit circle shown in .
We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. See .
First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. See . Point 
       is at an angle 
       from the positive x-axis with coordinates 
       and point 
       is at an angle of 
       from the positive x-axis with coordinates 
       Note the measure of angle 
       is 
      
    
Label two more points: 
       at an angle of 
       from the positive x-axis with coordinates 
       and point 
       with coordinates 
       Triangle 
       is a rotation of triangle 
       and thus the distance from 
       to 
       is the same as the distance from 
       to 
      
    
We can find the distance from 
       to 
       using the distance formula.
Then we apply the Pythagorean identity and simplify.
Similarly, using the distance formula we can find the distance from 
       to 
      
    
Applying the Pythagorean identity and simplifying we get:
Because the two distances are the same, we set them equal to each other and simplify.
Finally we subtract 
       from both sides and divide both sides by 
      
    
Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.

### Using the Sum and Difference Formulas for Sine

The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.

### Using the Sum and Difference Formulas for Tangent

Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.
Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall, 
      
    
Let’s derive the sum formula for tangent.
We can derive the difference formula for tangent in a similar way.

### Using Sum and Difference Formulas for Cofunctions

Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall from Right Triangle Trigonometry that, if the sum of two positive angles is 
       those two angles are complements, and the sum of the two acute angles in a right triangle is 
       so they are also complements. In , notice that if one of the acute angles is labeled as 
       then the other acute angle must be labeled 
      
    
Notice also that 
       which is opposite over hypotenuse. Thus, when two angles are complementary, we can say that the sine of 
       equals the cofunction of the complement of 
       Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.
From these relationships, the cofunction identities are formed. Recall that you first encountered these identities in The Unit Circle: Sine and Cosine Functions.
Notice that the formulas in the table may also be justified algebraically using the sum and difference formulas. For example, using
we can write

### Using the Sum and Difference Formulas to Verify Identities

Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems. Reviewing the general rules presented earlier may help simplify the process of verifying an identity.

### Key Equations


### Key Concepts

1. The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.
2. The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle. See  and .
3. The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle. See .
4. The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. See .
5. The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the product of the tangents of the angles. See .
6. The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles. See .
7. The cofunction identities apply to complementary angles and pairs of reciprocal functions. See .
8. Sum and difference formulas are useful in verifying identities. See  and .
9. Application problems are often easier to solve by using sum and difference formulas. See  and .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the exact value.
For the following exercises, rewrite in terms of 
         and 
        
      
For the following exercises, simplify the given expression.
For the following exercises, find the requested information.
For the following exercises, find the exact value of each expression.

### Graphical

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.
For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 
        
        )

### Technology

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

### Extensions

For the following exercises, prove the identities provided.
For the following exercises, prove or disprove the statements.