Sum and Difference Formulas (With Proofs and Examples)
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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.
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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
x22 + y22 = 1
x32 + 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 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 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.
- Find the exact value of sin (α + β).
- Find the actual value of tan (α + β).
- 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.
- cos (75°)
- 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