MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2022
P1
Trigonometric identity
Trigonometric identity
Source: Russian TST 2022, Day 7 P1
March 21, 2023
algebra
trigonometry
Problem Statement
Let
a
a{}
a
and
b
b{}
b
be positive integers. Prove that for any real number
x
x{}
x
∑
j
=
0
a
(
a
j
)
(
2
cos
(
(
2
j
−
a
)
x
)
)
b
=
∑
j
=
0
b
(
b
j
)
(
2
cos
(
(
2
j
−
b
)
x
)
)
a
.
\sum_{j=0}^a\binom{a}{j}\big(2\cos((2j-a)x)\big)^b=\sum_{j=0}^b\binom{b}{j}\big(2\cos((2j-b)x)\big)^a.
j
=
0
∑
a
(
j
a
)
(
2
cos
((
2
j
−
a
)
x
)
)
b
=
j
=
0
∑
b
(
j
b
)
(
2
cos
((
2
j
−
b
)
x
)
)
a
.
Back to Problems
View on AoPS