This page lists the formulae for tan nx for n=2, up to n=10. By studying the
table of coefficients of these polynomials, we can infer other formulae. For
instance, ,
relating the coefficients of the terms in the numerator (H) of the tan
nx formula (or numbers in the table) to the numerator and denominator
(K) of tan (n-1)x, thus enabling us to compute the coefficients of tan
nx from those of tan(n-1)x. Similarly, for the coefficients of the terms of the denominator.
See also:
The formula repeated from above are meant to clarify the table. The powers
are written, for instance, 1/0, where the top figure relates to the
power of the corresponding term in the numerator (top), and 0 relates
to the power of the corresponding coefficient of the term in the
denominator (bottom). The term number, k, is 0, 1, 2...
Power
1/0
(3/2
5/4
7/6
9/8
11/10
k
0
1
2
3
4
5
Formula
n
0
Top
0
.
tan(0)=0
Bottom
1
1
Top
1
tan(1x)=tan(x)
Bottom
1
2
Top
2
Bottom
1
-1
3
Top
3
-1
Bottom
1
-3
4
Top
4
-4
Bottom
1
-6
1
5
Top
5
-10
1
Bottom
1
-10
5
6
Top
6
-20
6
Bottom
1
-15
15
-1
7
Top
7
-35
21
-1
Bottom
1
-21
35
-7
8
Top
8
-56
56
-8
Bottom
1
-28
70
-28
1
9
Top
9
-84
126
-36
1
Bottom
1
-36
126
-84
9
10
Top
10
-120
252
-120
10
Bottom
1
-45
210
-210
45
-1
Observations
from the Table
What follows are observations, not all of which are proved on this
page. At this stage, they are hypotheses (if we are thinking
scientically about the data in the table) or conjectures (if we are
thinking mathematically). [They are all provable, however]
There are many patterns in the above table and formulae, some of which are mentioned below.
The first coefficient of the numerator (coefficient of tan
x) is always n. For instance,
when n=5:
the coefficient is 5
When n is even, the numerator
coefficients and the denominator
coefficients are symmetrical:
For instance, when n=10:
We notice the numerator coefficients are 10, -120, 252, -120,10, and the
denominator coefficients are: 1, -45,
210, -210, 45,
-1
The first coefficient in the denominator is always
1.
The signs alternate.
When n is odd, the last coefficient in the numerator is
±1. When n is even it is ±n
When n is even, the last coefficient in the denominator is
±1. When n is odd it is ±n.
The relationship between the coefficient of a term, n, k and the coefficient of term n-1, k is Where is the coefficient of the kth term of the nth
numerator, and is the coefficient of the kth term of the nth denominator. And
,
relating the coefficients of the terms in the numerator (H) of the tan
nx formula (or numbers in the table) to the numerator and denominator
(K) of tan (n-1)x, thus enabling us to compute the coefficients of tan
nx from those of tan(n-1)x. Similarly, 
for the coefficients of the terms of the denominator.
See also:
is the coefficient of the kth term of the nth
numerator, and 
is the coefficient of the kth term of the nth denominator. And