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<math display="block">T_n(\cos \theta) = \cos(n\theta).</math>
<math display="block">U_n(\cos \theta) \sin \theta = \sin\big((n + 1)\theta\big).</math>
The '''Chebyshev polynomials of the third kind''' <math>V_n</math> are defined by:
<math display="block">V_n(x)=\frac{\cos (n+1 / 2) \theta}{\cos (\theta / 2)} \text {, where } \quad x=\cos (\theta) \text.</math>
The '''Chebyshev polynomials of the fourth kind''' <math>W_n</math> are defined by:
<math display="block">W_n(x)=\frac{\sin (n+1 / 2) \theta}{\sin (\theta / 2)}, \quad \text { where } x=\cos (\theta).</math>
That these expressions define polynomials in <math>\cos\theta</math> may not be obvious at first sight but follows by rewriting <math>\cos(n\theta)</math> and <math>\sin\big((n+1)\theta\big)</math> using [[de Moivre's formula]] or by using the [[List of trigonometric identities#Angle sum and difference identities|angle sum formulas]] for <math>\cos</math> and <math>\sin</math> repeatedly. For example, the [[List of trigonometric identities#Double-angle formulae|double angle formulas]], which follow directly from the angle sum formulas, may be used to obtain <math>T_2(\cos\theta) = \cos(2\theta) = 2\cos^2\theta-1</math> and <math>U_1(\cos\theta)\sin\theta = \sin(2\theta) = 2\cos\theta\sin\theta</math>, which are respectively a polynomial in <math>\cos\theta</math> and a polynomial in <math>\cos\theta</math> multiplied by <math>\sin\theta</math>. Hence <math>T_2(x) = 2x^2 - 1</math> and <math>U_1(x) = 2x</math>.
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