Introduction to the Cosine Addition and Subtraction Formulas

The cosine addition and subtraction formulas are fundamental identities in trigonometry. They express the cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. These identities enable the simplification of complex trigonometric expressions and serve as building blocks for advanced topics such as Fourier analysis, wave mechanics, and complex analysis. The standard forms are:

Cosine Addition Formula: cos(A + B) = cos A cos B – sin A sin B

Cosine Subtraction Formula: cos(A – B) = cos A cos B + sin A sin B

Understanding the proofs of these formulas strengthens conceptual knowledge of trigonometry and reveals connections between geometry, algebra, and calculus. This article presents three rigorous proofs: a geometric proof using the unit circle, an algebraic proof based on the distance formula, and a succinct proof via Euler’s formula. Each approach offers a unique insight into why these identities hold, confirming their universal validity across mathematical domains.

Geometric Proof Using the Unit Circle

Derivation for Cosine Addition

Start with a unit circle centered at the origin O. Let point P correspond to angle A measured counterclockwise from the positive x-axis. The coordinates of P are (cos A, sin A). Similarly, let point Q correspond to angle B with coordinates (cos B, sin B). To find cos(A + B), consider rotating the standard coordinate axes by angle A so that the x-axis aligns with OP. In this rotated frame, the point that originally had angle B now has a net angle of A + B relative to the original x-axis.

The rotation can be expressed using a 2D rotation matrix. Rotating the point (cos B, sin B) by angle A gives:

R(A) = [ cos A   –sin A ]
          [ sin A       cos A ]

Multiplying the matrix by the column vector (cos B, sin B) yields:

x = cos A cos B – sin A sin B
y = sin A cos B + cos A sin B

Because the rotated point lies on the unit circle, its x-coordinate equals the cosine of the total angle A + B. Therefore:

cos(A + B) = cos A cos B – sin A sin B

This derivation illustrates how rotation transforms the problem into a coordinate geometry setting, making the addition formula visually intuitive.

Derivation for Cosine Subtraction

The subtraction formula follows directly from the addition formula by replacing B with –B. Using the even and odd properties of cosine and sine—cos(–B) = cos B and sin(–B) = –sin B—we compute:

cos(A – B) = cos(A + (–B)) = cos A cos(–B) – sin A sin(–B) = cos A cos B + sin A sin B

Thus, the subtraction formula emerges naturally from the symmetric properties of the unit circle. This approach reinforces that the sign change in the sine term is a direct consequence of sine’s oddness.

Algebraic Proof Using the Distance Formula

An alternative approach uses the Euclidean distance between two points on the unit circle. This method relies only on coordinate geometry and the law of cosines, making it purely algebraic. Let P and Q be points on the unit circle at angles A and B respectively.

  • Coordinates: P = (cos A, sin A), Q = (cos B, sin B)
  • Distance squared (by coordinate difference): (cos A – cos B)² + (sin A – sin B)²
  • Expand: cos²A – 2 cos A cos B + cos²B + sin²A – 2 sin A sin B + sin²B
  • Simplify using cos² + sin² = 1: (1) – 2 cos A cos B + (1) – 2 sin A sin B = 2 – 2(cos A cos B + sin A sin B)

Now consider triangle OPQ. The angle at the center between vectors OP and OQ is |A – B|. Applying the law of cosines to side PQ gives:

PQ² = OP² + OQ² – 2·OP·OQ·cos(A – B) = 1 + 1 – 2 cos(A – B) = 2 – 2 cos(A – B)

Equating the two expressions for PQ²:

2 – 2(cos A cos B + sin A sin B) = 2 – 2 cos(A – B)

Dividing both sides by –2 and canceling the common factor yields the subtraction formula. As with the geometric proof, replacing B with –B immediately produces the addition formula. This algebraic derivation confirms the identity without relying on rotation, using only basic algebra and the law of cosines.

Proof Using Euler’s Formula

Euler’s formula, eiθ = cos θ + i sin θ, links trigonometry to complex exponentials and provides an elegant proof. For real numbers A and B, consider the complex exponential of the sum:

ei(A+B) = eiA eiB = (cos A + i sin A)(cos B + i sin B)

Expanding the product using the distributive property and recalling that i² = –1:

= cos A cos B + i cos A sin B + i sin A cos B + i² sin A sin B
= (cos A cos B – sin A sin B) + i (sin A cos B + cos A sin B)

But by Euler’s formula, ei(A+B) also equals cos(A+B) + i sin(A+B). Equating the real parts of both expressions yields the addition formula for cosine:

cos(A + B) = cos A cos B – sin A sin B

The subtraction formula follows by substituting B with –B and using the even/odd properties of cosine and sine. This proof elegantly demonstrates how the algebraic structure of complex numbers encapsulates trigonometric identities, offering a powerful tool for further exploration in complex analysis and differential equations.

Applications of the Cosine Addition and Subtraction Formulas

Solving Trigonometric Equations

These formulas are essential for simplifying equations that involve sums or differences of angles. For example, solving cos 2x + cos x = 0 becomes straightforward when cos 2x is rewritten as cos(x + x) = cos² x – sin² x. The identity reduces the equation to one in terms of a single trigonometric function, allowing exact solutions. Common techniques include converting sums into products using the derived product-to-sum formulas, which are direct corollaries of the addition and subtraction formulas.

Wave Superposition and Interference

In physics, the superposition of two waves of different frequencies produces beats and interference patterns. The sum of two cosine waves, cos(ω₁t) and cos(ω₂t), can be expressed using the identity:

cos α + cos β = 2 cos((α+β)/2) cos((α-β)/2)

This identity, derived from the subtraction formula, shows that the resultant wave has a slowly varying envelope (the difference frequency) modulating a fast carrier (the average frequency). Engineers use this relationship in amplitude modulation (AM) radio and in designing mixers for frequency conversion. The same principle underlies the phenomenon of beats in acoustics, where two slightly different tuning forks produce a rhythmic variation in loudness.

Fourier Series and Signal Processing

Fourier series represent periodic signals as infinite sums of sines and cosines. The orthogonality of these basis functions—proved using the product-to-sum identities derived from the addition formulas—enables efficient decomposition of signals. The discrete cosine transform (DCT), widely used in image and audio compression (JPEG, MP3), directly exploits cosine addition properties. By breaking a signal into its frequency components, the DCT reduces redundancy and enables lossy compression with minimal perceptual distortion. For further details, see the Wikipedia article on the Discrete Cosine Transform.

Phase Shifts in Electrical Engineering

In alternating current (AC) circuit analysis, voltage and current are often out of phase. The cosine addition formulas allow engineers to combine sinusoidal functions with different phase angles. For example, the sum of two voltages with the same frequency but different phases can be expressed as a single cosine with a magnitude and phase shift determined by the addition identity. This simplifies the analysis of power systems and filter design. A detailed treatment of this application is available in All About Circuits’ guide to AC wave operations.

The law of cosines, which generalizes the Pythagorean theorem to any triangle, is derived from the cosine subtraction formula. Navigators use the law of cosines in spherical trigonometry to compute great-circle distances between points on Earth. Astronomers apply the same identities to calculate angular separations between celestial objects. These practical applications underscore the importance of the addition and subtraction formulas beyond pure mathematics.

Summary of Proofs and Connections

The cosine addition and subtraction formulas are not isolated facts; they emerge naturally from the geometry of the unit circle, from coordinate algebra using distance, and from the algebraic properties of complex exponentials. The geometric proof offers visual intuition, the algebraic proof relies on Euclidean distance and the law of cosines, and Euler’s formula reveals a deep link between trigonometry and complex analysis. Each proof validates the same result, reinforcing the consistency of mathematics.

For those seeking further study, Khan Academy’s Trigonometry section provides interactive practice, while MathWorld’s page on trigonometric addition formulas offers a comprehensive reference. Mastery of these proofs equips students and professionals with a firm foundation for advanced work in science, engineering, and mathematics.