DERIVATION OF FUNDAMENTAL TRIGONOMETRIC IDENTITIES

In this article, the step-by-step derivation of the fundamental trigonometric identities is analyzed. Trigonometry is one of the most essential branches of mathematics and has significant practical applications in fields such as geometry, physics, engineering, and computer graphics. By examining the relationships among the identities, the geometric meaning of the sine, cosine, tangent, and cotangent functions, and the fundamental connections between the trigonometric ratios of an angle, the main formulas are proven. The article also discusses the derivation of identities using the unit-circle approach, similarity principles, the Pythagorean theorem, sum and difference formulas, even-odd properties, and trigonometric expressions involving products and sums. The results of the study provide a deeper understanding of how the fundamental formulas—widely applied in trigonometric equations, analytic geometry, and the theory of functions—are derived.

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