Sum and Difference Angle Formula Proofs

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Sum and Difference Angle Formula Proofs

Last Update: October 7,2025

Introduction
Right Triangle Trig Functions
Two Right Triangles
Define the Sides
Sin and Cos Angle Sum Formulas
Sin and Cos Angle Difference Formulas
Tan Angle Formulas
Summary of Sin, Cos, Tan Angle Formulas

Introduction

In this post, I derive the sine, cosine, and tangent sum and difference angle formulas.

The sum and difference angle formulas are some of the most powerful tools in trigonometry.

While they’re often used to find the exact values for unusual angles like 75∘ or 15∘, their deeper power lies in rewriting expressions—a crucial skill in higher mathematics.

These formulas directly lead to the double-angle identities, which in turn are algebraically rearranged to create the vital Power-Reduction Formulas.

For instance, the identity cos²(x) =1/2(1+cos(2x)) reduces the square of a cosine to a linear form, making it significantly easier to integrate in Calculus.

Sum and difference angle formulas are also used to

derive the half-angle formulas,
verify complex trigonometric identities in proofs, and
solve a wide variety of trigonometric equations.

The Equations

sin(α + β) = sinα cosβ + cosα sinβ
sin(α – β) = sinα cosβ – cosα sinβ
cos(α + β) = cosα cosβ – sinα sinβ
cos(α – β) = cosα cosβ + sinα sinβ
tan(α + β) = (tanα + tanβ) / (1 – tanα tanβ)
tan(α – β) = (tanα – tanβ) / (1 + tanα tanβ)

To memorize the sine and cosine identities think of them as follows:

sin(α ± β) = sinα cosβ ± cosα sinβ

cos(α ± β) = cosα cosβ ∓ sinα sinβ

Good References

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Trigonometry Review

We’ll need to remember some trigonometry so here is your relevant review.

Refer to my post Geometry and Trigonometry Rules for more details.

Right Triangle Rules

For a right triangle as shown below:

Pythagorean Theorem: a² + b² = c²
sin²(θ) + cos²(θ) = 1 ; Pythagorean Identity
sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent = (hypotenuse)(sin(θ))/(hypotenuse)(cos(θ)) = sin(θ)/cos(θ)
mnemonic: SohCahToa

Even-Odd Identities (Identities for Negative Angles)

cos(−a) = cos(a) ; The cosine function is even.

The cosine function is even because the sign of the angle doesn’t change the value of the function.

sin(−a)=−sin(a); The sine function is odd.

The sine function is odd because it is symmetric about the origin, meaning the function value flips its sign.

tan(−a)=−tan(a); The tangent function is odd like the sine function

The tangent function is odd because it is a quotient of an odd and an even function (sin(a)/cos(a))

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Two Right Triangles

The proof for the sum and difference angle formulas comes from constructing two right triangles and doing some basic trig on them.

Consider the picture below as you read the set up steps.

1. Draw a right Triangle abc

2. Draw another right Triangle acd whose base is the hypotenuse of the Triangle abc

3. Designate angle cab (∠cab) as α and ∠dac as β, and ∠dab as (α+β)

4. Let the hypotenuse of Triangle acd = 1

Picture: Right Triangle Set Up for Proving Sum and Difference Angle Formulas

5. Draw perpendicular lines ce and ed such that e is at the same “height” as d.

6. Do the same to draw lines af and fd.

7. Since line ad intersects parallel lines ab and fe , their alternate interior angles are equal.

8. When a line intersects two parallel lines, the alternate interior angles are equal. Since line ad intersects parallel lines fe and ab, then angles ∠dab = α+β = ∠fda.

9. 180 degrees = ∠cab + ∠abc + ∠bca = α + 90 + ∠bca; so ∠bca = 90 – α

10. 180 degrees = ∠bca + ∠acd + ∠dce = 90 – α + 90 + ∠dce; so ∠dce = α

We can define all the sides now.

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Use Right Triangle Trig Identities to Define the sides

We have the same picture below.

We can apply basic trig rules now to define the sides:

Picture: Right Triangle Set Up for Proving Sum and Difference Angle Formulas

11. Since Triangle Δacd is a right triangle with a hypotenuse of 1, then:

11.1 sinβ = opp/hyp → dc = sinβ

11.1 cosβ = adj/hyp → ac = cosβ

12. Since Triangle Δabc is a right triangle with a hypotenuse of cosβ, then:

12.1 sinα = opp/hyp → bc = sinαcosβ

12.1 cosα = adj/hyp → ab = cosαcosβ

13. Since Triangle Δdec is a right triangle with a hypotenuse of sinβ, then:

13.1 sinα = opp/hyp → de = sinαsinβ

13.2 cosα = adj/hyp → ec = cosαsinβ

14. Since Triangle Δdfa is a right triangle with a hypotenuse of 1, then:

14.1 sin(α+β) = opp/hyp → af = sin(α+β)

14.2 cos(α+β) = adj/hyp → df = cos(α+β)

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Sin and Cos Angle Addition (Sum) Formulas

We’ll bring the same picture down here again.

Picture: Right Triangle Set Up for Proving Sum and Difference Angle Formulas

This rectangle has equal sides, so line fe = line ab and line fa = line eb, so

15. fe = fd + de = ab → cos(α+β) + sinαsinβ = cosαcosβ

15.1 cos(α+β) = cosαcosβ – sinαsinβ ; Cosine Angle Addition Formula

16. and fa = eb = ec + cb → sin(α+β) = cosαsinβ + sinαcosβ

16.1 sin(α+β) = cosαsinβ + sinαcosβ ; Sine Angle Addition Formula

Sin and Cos Angle Subtraction (Difference) Formulas

Let’s get the difference versions of 15.1 and 16.1

In 15.1 we can substitute -β for β:

17. cos(α+-β) = cosαcos(-β) – sinαsin(-β)

but cos(-β) = cos(β) and sin(-β) = -sin(β) , so

17.1 cos(α-β) = cosαcosβ + sinαsinβ; Cosine Angle Subtraction (Difference) Formula

In 16.1 we can substitute -β for β:

18. sin(α+-β) = cosαsin(-β) + sinαcos(-β)

but cos(-β) = cos(β) and sin(-β) = -sin(β) , so

18.1 sin(α-β) = sinαcos(β) – cosαsin(β); Sine Angle Subtraction (Difference) Formula

Tan Angle Addition (Sum) and Subtraction (Difference) Formulas

From the tangent trig identity, we know that,

19. tan(α+β) = opposite/adjacent = sin(α+β)/cos(α+β)

so substituting equations (17.1) and (18.1) into (19) we get,

20. tan(α+β) = (cosαsinβ + sinαcosβ)/(cosαcosβ – sinαsinβ)

Divide the Numerator and Denominator by cosαcosβ:

21. tan(α+β) = (cosαsinβ/cosαcosβ + sinαcosβ/cosαcosβ)/(cosαcosβ/cosαcosβ – sinαsinβ/cosαcosβ)

The four terms in equation (21) simplify neatly:

21.1 cosαsinβ/cosαcosβ = (1)(tanβ)

21.2 sinαcosβ/cosαcosβ = (tanα)(1)

21.3 cosαcosβ/cosαcosβ = 1

21.4 sinαsinβ/cosαcosβ = tanαtanβ

so (21) becomes

22. tan(α+β) = ((1)(tanβ) + (tanα)(1))/(1 – tanαtanβ)

23. tan(α+β) = (tanβ + tanα)/(1 – tanαtanβ); Tangent Angle Addition (Sum) Formula

We can get the difference version of equation 23 by repeating what we did for the sin and cos expressions.

Substitute -β for β in (23):

24. tan(α+-β) = ((tan(-β)) + (tanα))/(1 – tanαtan(-β))

Like sin, tan is an odd function, which means $tan (-β) = - tan (β), so$

25. tan(α-β) = ((-tan(β)) + (tanα))/(1 + tanαtan(β))

25. tan(α-β) = (tanα-tanβ)/(1 + tanαtanβ) ; Tangent Angle Subtraction (Difference) Formula

Example: cos²(x) and sin²(x)

The derivation of the Root Mean Square (RMS) of a sinusoidal function requires solving the integral of cos²(x) (or sin²(x)).

The integration simplifies when we replace

∫cos²( $ω$ t)dt with its equality ∫1/2(1+ cos(2 $ω$ t))dt

∫sin²( $ω$ t)dt with its equality ∫1/2(1 – cos(2 $ω$ t))dt

See this great video to learn more about this RMS example : Ch 12 AC Power (23 of 58) How to Calculate the RMS Current? Michel van Biezen

We can show that

cos²( $ω$ t) equals 1/2(1+ cos(2 $ω$ t))

and

sin²( $ω$ t) equals 1/2(1- cos(2 $ω$ t))

by using the cosine angle summation identity.

15.1 cos(α+β) = cosαcosβ – sinαsinβ ; Cos Angle Addition Formula

So, let’s show that cos²( $ω$ t) = 1/2(1+ cos(2 $ω$ t)).

Start with cos(2x) and apply equation 15.1:

e1. cos(2x) = cos(x + x) = cosxcosx – sinxsinx = cos²(x) – sin²(x)

We also know (Pythagorean identity; see my Geometry and Trigonometry Rules post) that :

e2. sin²(x) + cos²(x) = 1 ; Pythagorean Identity

e3.1 sin²(x) = 1 – cos2(x)

e3.2 cos2(x) = 1 – sin²(x)

Substitute using 3.1 or 3.2 (sin²(x) or cos2(x)) in the cos(2x) expression (equation 1.)

e4.1 cos(2x) = cos²(x) – (1 – cos2(x)) = 2cos2(x) – 1

e4.2 cos(2x) = 1 – sin²(x) – sin²(x) = 1 – 2sin2(x)

solving for cos2(x) we get,

e5.1 cos²(x) = 1/2(1+ cos(2x))

e5.2 sin²(x) = 1/2(1 – cos(2x))

Let’s integrate ∫cos²( $ω$ t)dt using (e5.1):

∫cos²( $ω$ t)dt

∫1/2(1+ cos(2ωt))dt

Separate the terms and apply the basic rules of integration.

Separate the Constant and Terms:

\int1/2 (1 + cos (2 ω t)) d t =1/2 \int 1 d t +1/2 \int cos (2 ω t) d t

Integrate the First Term:

\int 1 d t =1/2 t

Integrate the Second Term (using Substitution):

For $\int cos (2 ω t) d t$ , let $u = 2 ω t$

Then the derivative is

$d u = 2 ω d t$

This means $d t =(1/(2 ω)) d u$

Substitute into the integral:

$\int cos (u) (1/(2 ω) d u) =1/2 \cdot1/(2 ω) \int cos (u) d u$ $=1/(4 ω) sin (u)$
Substitute back $u = 2 ω t$ : $1/(4 ω sin (u) = 1/(4$ ω)sin(2ωt)

Solution:

\int1/2(1+ cos(2 ω t))dt = (1/2) t + 1/(4 ω) sin (2 ω t)

Let’s integrate ∫sin²( $ω$ t)dt using (e5.2):

∫sin²( $ω$ t)dt

∫1/2(1 – cos(2 $ω$ t))dt

Separate the terms and apply the basic rules of integration.

Separate the Constant and Terms:

\int1/2 (1- cos (2 ω t)) d t =1/2 \int 1 d t- 1/2 \int cos (2 ω t) d t

Integrate the First Term:

\int 1 d t =(1/2) t

Integrate the Second Term (using Substitution):

For $\int cos (2 ω t) d t$ , let $u = 2 ω t$

Then the derivative is

$d u = 2 ω d t$

This means $d t =(1/(2 ω)) d u$

Substitute into the integral:

$\int cos (u) (1/(2 ω) d u) =1/2 \cdot1/(2 ω) \int cos (u) d u$ $=1/(4 ω) sin (u)$
Substitute back $u = 2 ω t$ : $1/(4 ω sin (u) = 1/(4$ ω)sin(2ωt)

Solution:

\int1/2(1 - cos(2 ω t))dt = (1/2) t - 1/(4 ω) sin (2 ω t)

Summary – Sin, Cos, Tan Sum and Difference Angle Formulas

cos(α+β) = cosαcosβ – sinαsinβ ; Cosine Angle Addition Formula

sin(α+β) = sinαcosβ + cosαsinβ ; Sine Angle Addition Formula

cos(α-β) = cosαcosβ + sinαsinβ ; Cosine Angle Subtraction (Difference) Formula

sin(α-β) = sinαcos(β) – cosαsin(β) ; Sine Angle Subtraction (Difference) Formula

tan(α+β) = (tanα + tanβ)/(1 – tanαtanβ) ; Tangent Angle Addition (Sum) Formula

tan(α-β) = (tanα-tanβ)/(1 + tanαtanβ) ; Tangent Angle Subtraction (Difference) Formula

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Introduction

The Equations

Good References

Trigonometry Review

Right Triangle Rules

For a right triangle as shown below:

Pythagorean Theorem: a2 + b2 = c2

sin2(θ) + cos2(θ) = 1 ; Pythagorean Identity

sin(θ) = opposite/hypotenuse

cos(θ) = adjacent/hypotenuse

tan(θ) = opposite/adjacent = (hypotenuse)(sin(θ))/(hypotenuse)(cos(θ)) = sin(θ)/cos(θ)

mnemonic: SohCahToa

Even-Odd Identities (Identities for Negative Angles)

cos(−a) = cos(a) ; The cosine function is even.

sin(−a)=−sin(a); The sine function is odd.

tan(−a)=−tan(a); The tangent function is odd like the sine function

Two Right Triangles

Picture: Right Triangle Set Up for Proving Sum and Difference Angle Formulas

<img loading="lazy" loading="lazy" decoding="async" class="alignnone size-full wp-image-34423" src="https://wymhacks.com/wp-content/uploads/2025/10/2025.04.21-Sum-and-Difference-Angle-Formula-Proof-.jpg" alt="" width="696" height="682" />

Use Right Triangle Trig Identities to Define the sides

Picture: Right Triangle Set Up for Proving Sum and Difference Angle Formulas

<img loading="lazy" loading="lazy" decoding="async" class="alignnone size-full wp-image-34423" src="https://wymhacks.com/wp-content/uploads/2025/10/2025.04.21-Sum-and-Difference-Angle-Formula-Proof-.jpg" alt="" width="696" height="682" />

Sin and Cos Angle Addition (Sum) Formulas

Picture: Right Triangle Set Up for Proving Sum and Difference Angle Formulas

<img loading="lazy" loading="lazy" decoding="async" class="alignnone size-full wp-image-34423" src="https://wymhacks.com/wp-content/uploads/2025/10/2025.04.21-Sum-and-Difference-Angle-Formula-Proof-.jpg" alt="" width="696" height="682" />

15.1 cos(α+β) = cosαcosβ – sinαsinβ ; Cosine Angle Addition Formula

16.1 sin(α+β) = cosαsinβ + sinαcosβ ; Sine Angle Addition Formula

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Sin and Cos Angle Subtraction (Difference) Formulas

17.1 cos(α-β) = cosαcosβ + sinαsinβ; Cosine Angle Subtraction (Difference) Formula

18.1 sin(α-β) = sinαcos(β) – cosαsin(β); Sine Angle Subtraction (Difference) Formula

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Tan Angle Addition (Sum) and Subtraction (Difference) Formulas

23. tan(α+β) = (tanβ + tanα)/(1 – tanαtanβ); Tangent Angle Addition (Sum) Formula

25. tan(α-β) = (tanα-tanβ)/(1 + tanαtanβ) ; Tangent Angle Subtraction (Difference) Formula

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Example: cos2(x) and sin2(x)

15.1 cos(α+β) = cosαcosβ – sinαsinβ ; Cos Angle Addition Formula

e2. sin2(x) + cos2(x) = 1 ; Pythagorean Identity

e5.1 cos2(x) = 1/2(1+ cos(2x))

e5.2 sin2(x) = 1/2(1 – cos(2x))

Let’s integrate ∫cos2(ωt)dt using (e5.1):

∫1/2(1+ cos(2ωt))dt = (1/2)t + 1/(4ω)sin(2ωt)

Let’s integrate ∫sin2(ωt)dt using (e5.2):

∫1/2(1 – cos(2ωt))dt = (1/2)t – 1/(4ω)sin(2ωt)

Summary – Sin, Cos, Tan Sum and Difference Angle Formulas

cos(α+β) = cosαcosβ – sinαsinβ ; Cosine Angle Addition Formula

sin(α+β) = sinαcosβ + cosαsinβ ; Sine Angle Addition Formula

cos(α-β) = cosαcosβ + sinαsinβ ; Cosine Angle Subtraction (Difference) Formula

sin(α-β) = sinαcos(β) – cosαsin(β) ; Sine Angle Subtraction (Difference) Formula

tan(α+β) = (tanα + tanβ)/(1 – tanαtanβ) ; Tangent Angle Addition (Sum) Formula

tan(α-β) = (tanα-tanβ)/(1 + tanαtanβ) ; Tangent Angle Subtraction (Difference) Formula

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Pythagorean Theorem: a² + b² = c²

sin²(θ) + cos²(θ) = 1 ; Pythagorean Identity

Example: cos²(x) and sin²(x)

e2. sin²(x) + cos²(x) = 1 ; Pythagorean Identity

e5.1 cos²(x) = 1/2(1+ cos(2x))

e5.2 sin²(x) = 1/2(1 – cos(2x))

Let’s integrate ∫cos²( $ω$ t)dt using (e5.1):

$\int1/2(1+ cos(2 ω t))dt = (1/2) t + 1/(4 ω) sin (2 ω t)$

Let’s integrate ∫sin²( $ω$ t)dt using (e5.2):

$\int1/2(1 - cos(2 ω t))dt = (1/2) t - 1/(4 ω) sin (2 ω t)$