Addition and Subtraction Formulas for Sine and Cosine II

The simplest, truly "proofs without words" for the addition and subtraction formulas for sine and cosine have been published by R. Nelsen. As the other proofs relying on visual imagery, Nelsen's too are subject to natural argument (angle) limitations:

Formulas for sin(α + β) and cos(α + β)

	sin(α + β) = sinα cos β + cos α sin β.
	cos(α + β) = cosα cos β - sin α sin β.

Formulas for sin(α - β) and cos(α - β)

	sin(α - β) = sinα cos β - cos α sin β.
	cos(α - β) = cosα cos β + sin α sin β.

References

1. R. B. Nelsen, Proofs Without Words II, MAA, 2000, p. 46

Trigonometry

• What Is Trigonometry?
• Addition and Subtraction Formulas for Sine and Cosine
  • Sine of a Sum Formula
  • Addition and Subtraction Formulas for Sine and Cosine II
  • Addition and Subtraction Formulas for Sine and Cosine III
  • Addition and Subtraction Formulas
• The Law of Cosines (Cosine Rule)
• Cosine of 36 degrees
• Sine and Cosine of 15 Degrees Angle
• Sine, Cosine, and Ptolemy's Theorem
• arctan(1) + arctan(2) + arctan(3) = π
• arctan(1/2) + arctan(1/3) = arctan(1)
• Morley's Miracle
• Napoleon's Theorem
• A Trigonometric Solution to a Difficult Sangaku Problem
• Trigonometric Form of Complex Numbers
• Derivatives of Sine and Cosine
• ΔABC is right iff sin²A + sin²B + sin²C = 2
• Hunting Right Angles
• Point on Bisector in Right Angle
• Trigonometric Identities with Arctangents
• The Concurrency of the Altitudes in a Triangle - Trigonometric Proof

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