# 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.

### DETAILS

Reference
 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.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

• Wallis FormulaMichael Schreiber
• Continued Fraction Approximations of the Tangent FunctionMichael Trott
• Basis for Pure Geodetic AnglesIzidor Hafner
• Using Bernoulli's Formula to Sum Powers of the Integers from 1 to nEd Pegg Jr
• Special Case of Vandermonde's IdentityKevin Ren and Ananth Rao
• Factoring the Even Trigonometric Polynomials of A269254Brad Klee
• A Visual Proof of the Double-Angle Formula for SineChris Boucher
• Fry's Geometric Demonstration of the Sum of CubesS. M. Blinder
• Semi-Fibonacci PartitionsGeorge Beck
• Generalized Fibonacci Sequence and the Golden RatioS. M. Blinder