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Binomial Coefficients in the Multiple-Angle Tangent Formula

​
n:
1
2
3
4
5
6
7
8
9
tan(x)
tan(x)
1
1
1
1
Multiple-angle formulas for
tan(nx)
are found to contain binomial coefficients multiplying the powers
m
tan
(x)
, with odd
m
in the numerator and even
m
in the denominator. The binomial coefficients, corresponding to the numbers of the
th
n
row of Pascal's triangle, occur in the expression in a zigzag pattern (i.e. coefficients at positions
1,3,5,…
are in the denominator, coefficients at the positions
2,4,6,…
are in the numerator, or vice versa), following the binomials in row
n
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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