Internat. J. Math. & Math. Sci.
VOL. 12 NO.
(1989) 119-122
119
ON BAICA’S TRIGONOMETRIC IDENTITY
R.J. GREGORAC
Department of Mathematics
Iowa State University
Ames, Iowa 50011
(Received May 21, 1987 and in revised form September 7, 1987)
ABSTRACT:
The author obtains three new elementary trigonometri’c identities
including
(pql) (p-2..)
p-2
sin p:0
sin 8"
8
approach
pp-2
2
which, letting
p-I
2
2
0,
E
k=l
2k
P
28
p-l-k
cos-
gives a simple derivation of a recent result of M.
.Baica [I],
(p-l) (p-2)
p-2
2
(I
cos
k=l
KEY WORDS AND PHRASES.
-
(cos
2
k
p-l-k
Trigonometric identities, Chebyshev polynomials.
1980 AMS SUBJECT CLASSIFICATION CODE.
12E12, 42A05.
INTRODUCTION
Some interesting combinatorial relations that are difficult to prove directly
can be easily derived as limiting cases of simpler relations.
We here show that the
interesting recent result of Balca [I],
(p-l) (p-2)
pp-2
p-2
2
2
n (:
o
k=l
for
p > 3,
(1.3) below.
2kP-l-k
-7-
(.)
can be proved this way, by proving the elementary relations
Then (I.I) follows from (1.3) by letting
sides of (1.3).
(Since the
p-I
I,
8
approach
0
on both
the product in (1.3) can be
terms.)
p-2
considered as having only
(1.3) is
term of
(1.2) and
The identities
[sin pSI
2
2P-I
sin 8
Ps
n----E
sin 0
p > 2,
E
(cos
28
(1.2)
cos
k=l
and
for all
p-I
2
2
(p-1)(p-2) p-I
2
E (cos 20
k=l
cos
2k]
P
p-l-k
(1.3)
can be easily proved in several ways.
Proofs using polynomials seem simplest and because Chebyshev polynomials (of
the second kind)
U
p-I
(x)
sinlp
will be used in the arguments here.
cos-l(x))/sinlcos -I (x))
are well known
they
120
R.J. GREGORAC
U p-I (x)
The only properties of
p-I
polynomial of degree
cos(k/p),
....
k-
used here are the facts that
with leading coefficient
,p-l,
(-I,I).
in
2
p-I
U
p-I
(x)
is a
and zeros
For completeness we will prove these facts
in the next section.
CHEBYSHEV POLYNOMIALS.
2.
The Chebyshev polynomials of the first kind are defined by
P
l(X>l-x
T (x) + i U
(x
cos-l(x))."
cos(p
T (x)
)P.
+
(x
2
2
cos
p-I,
pe +
i
2 p.
Up_l(X)
k
cos---
3.
PROOFS OF (1.2) AND (1.3).
[cos
(2 p-l
p-I
R
U
T (x)
and
p
2
.(p-k)
cos-
p-I
(x)
2
U
I.
p
Tp+l(x)
+ i
sin
0) p
T (x) and U
(x)
p
p-I
both have leading
(x)
p-I
By expanding
U (x)
P
and
p-I
have leading
degree polynomial
can be directly verified in the definition
in terms of its roots
2
and by pairing terms in increasing and
2(p-I) p-I
ff
(x-
2(p-I)
Pl(2x2
(p-k)
).
P
It follows
2k]
-cos---./2
k-I
(3.1)
to see that
sin pc)
sin
cos
Ix
2
cos {)
)(x
and a double angle formula that
U2
p-I
x
cos
k-1
cos--P
P
2
p-I (x)
)
Ix- cos
U
decreasing order,
Now let
{)
given in the introduction.
Writing the polynomial
from
pe
i sin
l,...,p-l,
U
U2p-l(X)
then
it is seen that
k
P
(x)
(x),
Finally, that the roots of the
of
p-I
-I
this being clear for
coefficients
are
cos
Inductively assume
l-x2 )P (x + l-x2)
+ i
0
Expanding thls last expression shows
are polynomials.
coefficients
If
2
22(p-l)
e
p-l(cos_
II
2{)
cos
2
2k
.---1
proving (1.2).
Now denote
to the power
2x2-I
by
T2(x)
in equation (3.1) and raise both sides of (3.1)
p-2:
(Up_
(x)
,2(p-2)
p-1
2(P-l)(p-2)( II (T2(x)
k=l
cos
2tk]
----.)
p-2
Rewriting the product on the right in ascending and descending order and noting
that
cos
2k
P
cos
2’(p-k)
P
gives
BAICA’S TRIGONOMETRIC IDENTITY
(Up_
121
(x)) 2 (p-2)
2(P-l)(p-2)
p-I
p-I
2k)P-l-k
(T2(x)
k=l
(T2(x)
k=l
cos-
P
p-I
2(P-I)(p-2)( [Tm(X)
k=l
2w(p-k)]
P
p-l-k
2
l-k
2k]pP
-cos
cos
Thus
(p-
+/-(Up_l(x))P-2
(p--_2) p-1
2
2
.T2(x)
k=l
2k]
P
cos
p-l-k
Since this is a polynomial identity, the algebraic sign can be determined by
evaluating it at any
Then
lim
x
U
p-I
(x)
sin p0
sin 0
lim
0
0
must clearly be chosen.
x
Taking
T2(1)
and
p,
I,
proves (1.3).
cos 0
equation (I.I) is the limiting case of (1.3) at
4.
Let
which gives non-zero values.
x
0
x
approach
I.
so the positive sign
As noted earlier,
O.
FURTHER CONSEQUENCES OF (1.2).
The limiting case at
p
2
0
0
of (1.2) is
p-I
2P-I
(I
l[
2k].
P
cos
k=l
Now
sin20
sin
2 k
P
(l
cos 20)/2
(I
cos
2k
cos-
cos 20
2k]
/2
P
cos
2k
P
so, by (1.2) and the above limiting case of (1.2),
sin p0
sin
For odd
odd
p
2
p2
0
p-I
sin
(I
k=l
s
2
0
’’..P ).
(4.1)
this is the square of the known identity (see Spiegel
[2]) that for
p
sin pO
sin 0
2
sln2
(I
P
k:l
sln
2
k)O
(4.2)
P
thus generalizing (4.2) to all p. When p is even, the p/2 term in (4.1)
2
2
is
cos 0, so pleasing versions of (4.2) when p is even would be
sin 0
sin
sn
pO_ p cos 0
0
k=l
or
sin
[1
p0
sin 20
2
sin
I
(I
k=l
Finally, in view of the identity
sin
sin
sin
2
0
k
P
0
2 k
)"
(4.3)
P
(4.4)
R.J. GREGORAC
122
cos(i-J)
2 sin i sin
p-I
H (cos 20
p-I
II (cos 2k
2k
cos-’---
P
k=l
cos(i+j),
"--
kffil
p-I
(2 sin(e
+
sin(e
p-I
2P-l(k=l sln(0 +-))k
q
+
2
Thus by equation (1.2) we see that the known identity
sin p0
sin
0"
2p-I
p-I
R
k--I
k
+-1
sin(O
(4.5)
can be derived as well (Carlson [3]), or, conversely, can be used as the starting
point to prove (1.2) and (1.3).
We could not find a reference to an elementary proof of (4.5).
Many texts use
the gamma function (as in Carlson [3]) to derive the general formula and others only
’prove
0 (for example, see Spiegel [2]).
the limiting case 0
We conclude with the following direct proof of identity (4.5).
relation, sin p0
)/2i, so
(e Ip0 e
Note that, if
zP-I
of
z
e
210
are again all the roots of
zP-I.
2k
-i
z
the conjugates,
zP-l.
P
e
k
By Euler’s
of the ordinary roots of
Thus part of the above is a factorlzatlon
Computing the product gives
p(p.I)
i
p-I
k=O
k
’
(e210
(e ipO
e
-iPO(e
2
p
2i )P
e-ipO]i p-I
(21)P
sin pO
2
p-I
This proof of (4.5) is due to Professor James A. Wilson.
me to include it here.
1 thank him for allowing
REFERENCES
[l]
[2]
[3]
BAICA, MALVINA, Trigonometric Identities, Internat. J. Math. Math.
Sci. Vol.
9, No. 4 (1986), 705-714.
SPIEGEL, MURRAY R., Complex Variables, McGraw Hill, 1964, pp. 25 and 32.
CARLSON, B. C., Special Functions of Applied Mathematics, Academic Press
1977
p. 57.