Further Information

Ching-Kuang Shene (冼鏡光), Professor Emeritus
Department of Computer Science
Michigan Technological University
Houghton, MI 49931
USA
Created September 28, 2025
Updated October 21, 2025

This page provides you with more information on proofs of the angle sum difference and angle sum identities as well as other related proofs. These proofs did not use the Pythagorean Theorem and the Pythagorean Identity. For each publicationits publication year and a link (if available) to jstor.org are provided. Note that you need an account to download from jstor.org. If a publication addresses multiple categories, only one will category is chosen. Furthermore, none of the listed publications used the Pythagorean Theorem and the Pythagorean Identity; however, any one of them can be used to prove the Pythagorean Theorem and the Pythagorean Identity.

• Angle Difference Identities
  1. Sidney H. Kung (1989), Area and Difference Formulas
  2. Leonard M. Smiley (1999), Geometry of Subtraction Formulas
  3. Leonard M. Smiley (1999 or later), Geometry of Addition and Subtraction Formulas
• Angle Sum Identities
  1. Sidney H. Kung (1991), sin(x + y) = sin x cos y + cos x sin y for x + y < π
  2. Christopher Brueningsen (1993), sin(x + y) for x + y < π
  3. Sidney H. Kung (1995), cos(x + y) for x + y > π/2 (Check the last page, p. 145).
• The Double-Angle Identities
  1. Roger B. Nelsen (1989), The Double-Angle Formulas (You can use \( \cos(2x)=\cos^2(x)-\sin^2(x) \) to prove the Pythagorean Identity.)
• Other Interesting Results That Are Directly Related to the Pythagorean Theorem and Identity
  1. F. E. Wood (1949), Derivation of the Tangent Half-Angle Formula (This proof used the angle difference identity for sine to derive the half-angle formula from which you can get the Pythagorean Identity easily.)
  2. Stephen M. Robinson (1965), The Tangent of a Half-Angle (This proof did not use the angle difference identity for sine or cosine -- see F. E. Wood -- to derive the half-angle formula. Instead, it proves the following directly in a rather obvious way. Simplifying and rearranging the last two terms yields the Pytahgorean Identity immediately.) \[ \tan\left( \frac{\theta}{2} \right) = \frac{1-\cos(\theta)}{\sin(\theta)} = \frac{\sin(\theta)}{1+\cos(\theta)} \]
  3. Roger B. Nelsen (1989), The Substitution to Make a Rational Function of the Sine and Cosine (You can get the Pythagorean Identity Directly.)
• Calculus Based Proofs
  1. Mike Staring (1996), The Pythagorean Proposition: A Proof by Means of Calculus
• Books These books are in chronological order rather than alphabetical order. Click on the link, if there is one, to access the book on the web.
  1. Augustus de Morgan (1849), Trigonometry and Double Albegra
  2. Osmar Fort (1872), Lehrbuch der Analytischen Geometrie
  3. Ernest William Hobson (1891), A Treatise on Plane Trigonometry
  4. Friedrich Schur (1898), Lehrbuch der Analytischen Geometrie
  5. J. Versluys (1914), Zes en negentig bewijzen voor het theorema van Pythagora

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