Sum and Difference Formulas (With Proofs and Examples)

Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.

Sum and difference formulas are identities that involve trigonometric functions u + v or u - v for any angles of variables u and v. These formulas are significant for advanced work in mathematics. The sum and difference formulas are good identities used in finding exact values of sine, cosine, and tangent with angles that are separable into unique trigonometric angles (30°, 45°, 60°, and 90°). Shown below are the sum and difference identities for trigonometric functions.

Addition Formula for Cosine
cos (u + v) = cos (u) cos (v) - sin (u) sin (v)

Subtraction Formula for Cosine
cos (u - v) = cos (u) cos (v) + sin (u) sin (v)

Addition Formula for Sine
sin (u + v) = sin (u) cos (v) + cos (u) sin (v)

Subtraction Formula for Sine
sin (u - v) = sin (u) cos (v) - cos (u) sin (v)

Addition Formula for Tangent
tan (u + v) = [tan (u) + tan (v)] / [1 - tan (u) tan (v)]

Subtraction Formula for Tangent
tan (u - v) = [tan (u) - tan (v)] / [1 + tan (u) tan (v)]

Here are the proofs for the sum and difference formulas mentioned above.

Addition and Difference Formula for Cosine Proof

Let variables u and v be any real numbers. Let A be the point (1, 0), and then use u and v to locate the points B(x₁, y₁), C(x₂, y₂), and D(x₃, y₃) on the unit circle as indicated. For convenience, we assume 0 < v < u < 2π, though the results we get are valid for all actual values of variables u and v.

From the figure above, we note that arcs AC and BD have the same length. Hence, line segments AD and BD are also equal in size. Thus, by the distance formula, we have

AC = BD

√ [(x₂ - 1)² + (y₂ - 0)²] = √ (x₃ - x₁)² + (y₃ - y₁)²

By squaring both sides and removing the parentheses, we obtain the following equations.

x₂² - 2x₂ + 1 + y₂² = x₃² - 2x₁x₃ + x₁² + y₃² - 2y₁y₃ + y₁²

(x₂² + y₂²) + 1 - 2x₂ = (x₃² + y₃²) + (x₁² + y₁²) - 2x₁x₃ + 2y₁y₃

Since B, C, and D lie on the unit circle, it follows that:

x₁² + y₁² = 1

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x₂² + y₂² = 1

x^₃2 + y₃² = 1

Making these substitutions, we obtain the following equation.

1 + 1 - 2x₂ = 1 + 1 - 2x₁x₃ - 2y₁y₃

x₂ = x₃x₁ + y₃y₁

Finally, by substituting the values result to the following.

x₁ = cos (v)

x₂ = cos (u - v)

x₃ = cos (u)

y₁ = sin (v)

y₃ = sin (u)

x₂ = x₃ x₁ + y₃ y₁

cos (u - v) = cos (u) cos (v) + sin (u) sin (v)

The formula for cos (u + v) can be established by considering u + v =v - (-v) and substituting into the identity just derived. We obtain the following.

cos (u + v) = cos [u - (-v)]

cos (u + v) = cos (u) cos (-v) + sin (u) sin (-v)

Since cos (-θ) = cos (θ) and sin (-θ) = -sin (θ), thus gives cos (θ + θ) = cos (u) cos (v) - sin (u) sin (v).

Addition and Difference Formula for Sine Proof

We can prove the addition formula for sine using the cofunction identities and the subtraction formula for cosine.

sin (u + v) = cos [π/2 - (u + v)]
sin (u + v) = cos [(π/2 - u) - v]
sin (u + v) = cos (π/2 - u) cos (v) + sin (π/2 - u) sin (v)
sin (u + v) = sin (u) cos (v) + cos (u) sin (v)

Now, since u - v = u + (-v), it follows that:

sin (u - v) = sin [u + (-v)]
sin (u - v) = sin (u) cos (-v) + cos (u) sin (-v)
sin (u - v) = sin (u) cos (v) - cos (u) sin (v)

Addition and Difference Formula for Tangent Proof

In proving the addition formula for the tangent, let us use the primary identity for tangent wherein tangent is equal to the sine divided by cosine.

tan (u + v) = sin (u + v) / cos (u + v)
tan (u + v) = [sin (u) cos (v) + cos (u) sin (v)] / [cos (u) cos (v) - sin (u) sin (v)]

Divide the numerator and denominator by cos (u) cos (v), considering cos (u) cos (v) is not equal to zero.

tan (u + v) = [(sin (u) / cos(u)) (cos (v) / cos (v)) + ((cos (u) / cos (u)) (sin (v) / cos (v))] / [(cos (u) / cos (v)) (sin (u) / sin (v)) - (sin (u) / cos (u)) (sin (v) cos (v))]
tan (u + v) = [tan (u) + tan (v)] / [1 - tan (u) tan (v)]

The formula for tan (u - v) can be derived in the same manner as that for sin (u - v). To understand the sum and difference identities for all trigonometric equations, let us see the vast sum and difference formulas examples given below.

Example 1: Using the Subtraction Formula for Cosine

Find the exact value of cos (15°) by using the factual information that 15° = 60° - 45°.

Solution

Apply the subtraction formula of cosine. Substitute the following values directly to the equation.

u - v = 15°, u = 60°, and v = 45°

cos (u - v) = cos (u) cos (v) + sin (u) sin (v)

cos (15°) = cos (60°) cos (45°) + sin (60°) sin (45°)

cos (15°) = (½) (√2/2) + (√3/2) (√2/2)

cos (15°) = (√2 + √6) / 4

Final Answer

The exact value of cos (15°) using the subtraction formula for cosine is (√2 + √6) / 4.

Example 2: Using the Addition Formula for Cosine

Find the exact value of cos (7π/12) by using the general fact that 7π/12 = π/3 + π/4.

Solution

Simply apply the sum formula for cosine. Let variable u be π/3, and variable v be π/4.

cos (7π/12) = cos (π/3 + π/4)

cos (u + v) = cos (u) cos (v) - sin (u) sin (v)

cos (7π/12) = cos (π/3) cos (π/4) - sin (π/3) sin (π/4)

cos (7π/12) = (½) (√2 / 2) - (√3 / 2) (√2 / 2)

cos (7π/12) = [√2 - √6] / 4

Final Answer

The exact value of cos (7π/12) is [√2 - √6] / 4 or approximately -0.2588.

Example 3: Applying Both Sine and Cosine Sum and Difference Formulas

Find the exact value for sin (12°) cos (42°) - cos (12°) sin (42°) using the sine and cosine using the sum and difference formulas.

Solution

The given sine and cosine equation is a combination of functions that fits the difference formula for sine which is sin (u - v) = sin (u) cos (v) - cos (u) sin (v). This problem is just a reverse of the usual procedure. From the given equation, u = 12° and v = 42°.

sin (u - v) = sin (u) cos (v) - cos (u) sin (v)

sin (u) cos (v) - cos (u) sin (v) = sin (u - v)

sin (12°) cos (42°) - cos (12°) sin (42°) = sin (12° - 42°)

sin (12°) cos (42°) - cos (12°) sin (42°) = sin (-30°)

sin (12°) cos (42°) - cos (12°) sin (42°) = - sin (30°)

sin (12°) cos (42°) - cos (12°) sin (42°) = -1/2

Final Answer

The exact value for sin (12°) cos (42°) - cos (12°) sin (42°) is -½.

Example 4: Using the Addition Formula for Tangent

Find the exact value for the tangent equation (tan 80° + tan 55°) / (1 - tan 80° tan 55°).

Solution

The given tangent equation is a combination of functions that fits the sum formula for tangent which is tan (u + v) = [tan (u) + tan (v)] / [1 - tan (u) tan (v)].

[tan (u) + tan (v)] / [1 - tan (u) tan (v)] = tan (u + v)

(tan 80° + tan 55°) / (1 - tan 80° tan 55°) = tan (u + v)

(tan 80° + tan 55°) / (1 - tan 80° tan 55°) = tan (80° + 55°)

(tan 80° + tan 55°) / (1 - tan 80° tan 55°) = tan (135°)

tan (135°) = tan (180° - 45°)

(tan 80° + tan 55°) / (1 - tan 80° tan 55°) = -tan (45°)

(tan 80° + tan 55°) / (1 - tan 80° tan 55°) = -1

Final Answer

The exact value for the tangent equation (tan 80° + tan 55°) / (1 - tan 80° tan 55°) is -1.

Example 5:Using Multiple Sum and Difference Formulas

Establish the identity tan (x) + cot (y) = [cos (x - y)] / [cos (x) sin (y)].

Solution

This problem requires verification of identities using the addition and subtraction identities like sine and cosine, and some other basic identities. Apply the subtraction formula for cosine on the numerator of the given equation.

[cos (x - y)] / [cos (x) sin (y)] = [cos (x) cos (y) + sin (x) sin (y)] / cos (x) sin (y)

[cos (x - y)] / [cos (x) sin (y)] = [cos (x) cos (y) / cos (x) sin (y)] + [sin (x) sin (y) / cos (x) sin (y)]

[cos (x - y)] / [cos (x) sin (y)] = cot (y) + tan (x)

[cos (x - y)] / [cos (x) sin (y)] = tan (x) + cot (y)

Final Answer

The given equation tan (x) + cot (y) = [cos (x - y)] / [cos (x) sin (y)] is equal to tan (x) + cot (y).

Example 6: Subtraction Identity for Sine and Addition Identity for Cosine

Express sin (θ - 3π/2) and cos (θ + π) in terms of θ alone.

Solution

For sin (θ - 3π/2) and cos (θ + π), apply the subtraction formula for sine and addition formula for cosine, respectively.

Let u = θ and v = 3π/2.

sin (θ - 3π/2) = sin (u) cos (v) - cos (u) sin (v)

sin (θ - 3π/2) = sin (θ) cos (3π/2) - cos (θ) sin (3π/2)

sin (θ - 3π/2) = sin (θ) (0) - cos (θ) (-1)

sin (θ - 3π/2) = cos (θ)

cos (u + v) = cos (u) cos (v) - sin (u) sin (v)

cos (θ + π) = cos (θ) cos (π) - sin (θ) sin (π)

cos (θ + π) = cos (θ) (-1) - sin (θ) (0)

cos (θ + π) = - cos (θ)

Final Answer

The equations sine (θ - 3π/2) and cos (θ + π) are equal to cos (θ) and -cos (θ), respectively.

Example 6: Subtraction Identity for Sine and Addition Identity for Cosine

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Example 7: Simplifying a Complex Equation to a Single Term

Use the sum and difference identities to simplify the given expression to a single term.
cos (3x) cos (2y) + sin (3x) sin (2y)

Solution

To simplify the equation to a single term, apply the sum and difference identities. Let u = 3x and v = 2y.

cos (u) cos (v) + sin (u) sin (v) = cos (u - v)

cos (3x) cos (2y) + sin (3x) sin (2y) = cos (3x - 2y)

Final Answer

The simplified equation of cos (3x) cos (2y) + sin (3x) sin (2y) is cos (3x - 2y).

Example 8: Simplifying a Given Expression to a Single Term

Use the sum and difference identities to simplify the given expression to a single term.
sin (3) cos (1.2) - cos (3) sin (1.2)

Solution

The given expression applies the subtraction formula for sine. Simply let u = 3 and v = 1.2

sin (u - v) = sin (u) cos (v) - cos (u) sin (v)

sin (3) cos (1.2) - cos (3) sin (1.2) = sin (u - v)

sin (3) cos (1.2) - cos (3) sin (1.2) = sin (3 - 1.2)

sin (3) cos (1.2) - cos (3) sin (1.2) = sin (1.8)

Final Answer

The simplified expression of sin (3) cos (1.2) - cos (3) sine (1.2) is sin (1.8).

Example 9: Simplifying a Tangent Expression to a Single Term

Use the sum and difference identities to simplify the given expression to a single term.
[tan (325°) - tan (86°)] / [1 + tan (325°) tan (86°)]

Solution

The given expression applies the subtraction formula for the tangent. Simply let u = 325° and v = 86°.

tan (u - v) = [tan (u) - tan (v)] / [1 + tan (u) tan (v)]

[tan (325°) - tan (86°)] / [1 + tan (325°) tan (86°)] = tan (325° - 86°)

[tan (325°) - tan (86°)] / [1 + tan (325°) tan (86°)] = tan (239°)

Final Answer

By simplifying the given tangent expression, it results in tan (239°).

Example 10: Finding the Exact Values Using the Sum or Difference Identities in Trigonometry

Determine the exact value of the sine, cosine, and tangent of the angle 105° using the sum or difference identities.

Solution

For the angle measure 105°, let u = 60° and v = 45°.

sin (105°) = sin (60° + 45°)

sin (u + v) = sin (u) cos (v) + cos (u) sin (v)

sin (60° + 45°) = sin (60°) cos (45°) + cos (60°) sin (45°)

sin (60° + 45°) = (√3 / 2) (√2 / 2) + (½) (√2 / 2)

sin (60° + 45°) = (√6 + √2) / 4

cos (105°) = cos (60° + 45°)

cos (u + v) = cos (u) cos (v) - sin (u) sin (v)

cos (60° + 45°) = cos (60°) cos (45°) - sin (60°) sin (45°)

cos (60° + 45°) = (½) (√2 / 2) - (√3 / 2) (√2 / 2)

cos (60° + 45°) = - (√6 - √2) / 4

tan (105°) = tan (60° + 45°)

tan (u + v) = [tan (u) + tan (v)] / [1 - tan (u) tan (v)]

tan (60° + 45°) = [tan (u) + tan (v)] / [1 - tan (u) tan (v)]

tan (60° + 45°) = [tan (60) + tan (45)] / [1 - tan (60) tan (45)]

tan (60° + 45°) = (√3 + 1) / (1 - √3 * 1)

tan (60° + 45°) = -2 - √3

Final Answer

The final answers for sin (105°), cos (105°), and tan (105°) are (√6 + √2) / 4, - (√6 - √2) / 4, and -2 - √3, respectively.

Example 10: Finding the Exact Values Using the Sum or Difference Identities in Trigonometry

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Example 11: Verifying Identities

Express cos (θ - 3π/2) and sin (θ + 3π) in terms of θ alone.

Solution

By the difference identity for cos (u - v), we have the following solution. Let u = θ and v = 3π/2.

cos (u - v) = cos (u) cos (v) + sin (u) sin (v)

cos (θ - 3π/2) = cos (θ) cos (3π/2) + sin (θ) sin (3π/2)

cos (θ - 3π/2) = cos (θ) (0) + sin (θ) (-1)

cos (θ - 3π/2) = - sin (θ)

Similarly, using the identity for sin (u - v) gives the following equations.

sin (u + v) = sin (u) cos (v) + cos (u) sin (v)

sin (θ + 3π) = sin (θ) cos (3π) + cos (θ) sin (3π)

sin (θ + 3π) = sin (θ) (-1) + cos (θ) (0)

sin (θ + 3π) = - sin (θ)

Final Answer

The equations cos (θ - 3π/2) and sin (θ + 3π) are equal to -sin (θ).

Example 12: Using Addition Formulas to Find the Quadrant Containing an Angle

Suppose sin α = ⅘ and cos β = - 12/13, where α is in quadrant I and β in quadrant II.

1. Find the exact value of sin (α + β).
2. Find the actual value of tan (α + β).
3. Find the quadrant containing α + β.

Solution

In the illustration, angles α and β are shown. Since sin α = ⅘, we may choose the point (3,4) on the terminal side of α. Similarly, since cos β = - 12/13, the point (-12,5) is on the terminal side of β. Using the definition of the trigonometric functions of any angle, we have the following results.

sin α = ⅘

cos α = ⅗

tan α = 4/3

sin β = 5/13

cos β = - 12/13

tan β = -5/12

Substitute the obtained values to the addition formulas for sine and tangent.

sin (α + β) = sin (α) cos (β) + cos (α) sin (β)

sin (α + β) = (⅘) (- 12/13) + (⅗) (5/13)

sin (α + β) = -33 / 65

tan (α + β) = [tan (α) + tan (β)] / [1 - tan (α) tan (β)]

tan (α + β) = [(4/3) + (-5/12)] / [1 - (4/3)(-5/12)]

tan (α + β) = 33 / 56

Since sin (α + β) is negative and tan (α + β) is a positive value, the angle α + β must be in quadrant III.

Final Answer

The values of sin (α + β) and tan (α + β) are -33 / 65 and 33 / 56, respectively. The angle α + β is in quadrant III.

Example 13: Finding the Exact Values Using the Sum and Difference Identities for Cosine

Find the exact value for each of the following.

1. cos (75°)
2. cos (π/12)

Solution

In finding the exact value, we cannot use a calculator since it would produce a decimal approximation to the functional value. However, we may consider that 75° = 30° + 45°.

cos (u + v) = cos (u) cos (v) - sin (u) sin (v)

cos (75°) = cos (30° + 45°)

cos (30° + 45°) = cos (30°) cos (45°) - sin (30°) sin (45°)

cos (30° + 45°) = (√3 / 2) (√2 / 2) - (½) (√2 / 2)

cos (30° + 45°) = (√6 - √2) / 4

Similarly, we consider π/12 = π/3 - π/4 and apply the difference identity for cosine.

cos (u - v) = cos (u) cos (v) + sin (u) sin (v)

cos (π/12) = cos (π/3 - π/4)

cos (π/3 - π/4) = cos (π/3) cos (π/4) + sin (π/3) sin (π/4)

cos (π/3 - π/4) = (½) (√2 / 2) + (√3 / 2) (√2 / 2)

cos (π/3 - π/4) = (√6 + √2) / 4

Final Answer

The exact values of cos (75°) and cos (π/12) are (√6 - √2) / 4 and (√6 + √2) / 4, respectively.

Example 14: Finding the Exact Value of Cosine Given Multiple Conditions

Find the value of cos (u - v), given the following conditions.

• The variable u lies in quadrant III
• The variable v lies in quadrant I
• cos (u) = -15 / 17
• sin (v) = ⅘

Solution

Using the values cos (u) = -15 / 17 and sin (v) = ⅘, let us sketch angles u and v to obtain other values of sine and cosine. Take a look at the sketch.

cos (u) = -15/17

sin (u) = -8/17

cos (v) = ⅗

sin (v) = ⅘

cos (u - v) = cos (u) cos (v) + sin (u) sin (v)

cos (u - v) = (-15/17) (⅗) + (-8/17) (⅘)

cos (u - v) = -77 / 85

Final Answer

The value of cos (u - v) given the conditions stated earlier is -77 / 85.

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This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2021 Ray