Being an equality of complex numbers, one necessarily has equality both of the real parts and of the imaginary parts of both members of the equation. If "x", and therefore also formula_10 and formula_11, are real numbers, then the identity of these parts can be written (interchanging sides) as
These equations are in fact even valid for complex values of "x", because both sides are holomorphic functions of "x", and two such functions that coincide on the real axis necessarily coincide on the whole complex plane. Here are the concretre instances of these equations for formula_13 and formula_14:
The right hand side of the formula for formula_16 is in fact the value formula_17 of the Chebyshev polynomial formula_18 at formula_19

उपयोग
कोज्या (cosine) एवं ज्या (sine) के लिये सूत्र
गणितीय आगमन विधि से सिद्ध
उपपत्ति (Derivation)