## Sum and Difference of Angles Identities

Angle sum identities and angle difference identities can be used to find the function values of any angles however, the most feasible use of sum of angles trig identities is to identify the exact values of an angle that can be mathematically expressed as a sum or difference using the familiar values for the sine, cosine and tangent of the 30°, 45°, 60° and 90° angles as well their multiples.

The sum and difference formulas for sine and cosine are also helpful in dealing with the inverse trigonometric functions.

### Sum and Difference Angle Formulas

The sum formula for cosines implies that the cosine of the sum of two angles is equivalent to the product of the cosines of the angles subtracting the product of the sines of the angles. On the other hand, the difference formula for cosines implies that the cosine of the difference of two angles is equivalent to the product of the cosines of the angles adding up the product of the sines of the angles.

## Sum and Difference Angle Formulas

Sum of Angle Identities |
Difference of Angle Identities |

Sin [A + B] = sin A · cos B + cos A · sin B |
Sin [A – B] = sin A · cos B – cos A · sin B |

cos [A + B] = cos A · cos B – sin A · sin B |
Cos [A – B] = cos A · cos B + sin A · sin B |

tan [A + B] = tanA + tanB= 1- tanA· tan B |
tan [A – B] = tanA – tanB= 1+ tanA· tan B |

### Sum Formula for Tangent

The sum formula for tangent trigonometry implies that the tangent of the sum of two angles is equivalent to the sum of the tangents of the angles further divided by 1 minus (-) the product of the tangents of the angles.

### Difference Formula for Tangent

The Difference Formula for tangent implies that the tangent of the difference of two angles is equivalent to the difference of the tangents of the angles divided by 1 plus (+) the product of the tangents of the angles.

### Use of Pythagorean Theorem With the Sum and Difference Formulas

The Pythagorean Theorem when used along with the sum and difference formulas helps to find multiple sums and differences of angles.

## Solved Examples

### Example 1:

Identify the exact value of sine 15° applying the difference formula.

Solution:

Seeing that, 15° is the Value of Difference between 45° and 30°. Thus, the Sine Difference formula can be applied

Now, using the sin difference formula i.e. sin(A – B) = sin A · cos B – cos A · sin B. We get,

Sin [45° – 30°] = sin 45° · cos 30° – cos 45° · sin 30°

= √2/2. √3/2 -√2/2. √1/2

= √6/4 – √2/4 =√6-√2/4

Hence, sin 15° =√6-√2/4

### Example 2:

Identify the Exact value of cos 75° applying the sum formula.

Solution:

Seeing that, 75° is the sum of 45° and 30°. Thus, the cos sum formula can be applied

Now, using the cos sum formula i.e. cos (A + B) = cos A · cos B – sin A · sin B. We get,

cos (30° + 45°) = cos 30° · cos 30° · cos 45° – sin 30°· sin 45°

= √3/2. √2/2 -√1/2. √2/2

= √6/4 – √2/4 =√6-√2/4

Hence cos 75° =√6-√2/4

### Key Takeaways

1. You can have more than one option. For Example If you want to Find the sine of 120°

Sin (90 + 30) applying the Sum Formula

or

sin (180 – 60) applying the Difference Formula

Both Formulas Will Produce Similar Results.

2. In order to identify a secant, cosecant, or cotangent function, you would need to change the expression to one of the 3 basic functions, and perform the requisite calculations.

3. Remember to identify in which quadrant the angle will lie.