Binomial Coefficients in the Multiple-Angle Tangent Formula

Multiple-angle formulas for are found to contain binomial coefficients multiplying the powers , with odd in the numerator and even in the denominator. The binomial coefficients, corresponding to the numbers of the row of Pascal's triangle, occur in the expression in a zigzag pattern (i.e. coefficients at positions are in the denominator, coefficients at the positions are in the numerator, or vice versa), following the binomials in row of Pascal's triangle in the same order. In this Demonstration, corresponding coefficients are shown with the same color in both the tangent function expression and Pascal's triangle.

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Reference
[1] E. Maor, Trigonometric Delights, Princeton, NJ: Princeton University Press, 1998.
This was a project for Advanced Topics in Mathematics II, 2019–2020, Torrey Pines High School, San Diego, CA.
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