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Jozsef Wildt International Math Competition
2009 Jozsef Wildt International Math Competition
W. 20
Prove this trigonometric equation
Prove this trigonometric equation
Source: 2009 Jozsef Wildt International Mathematical Competition
April 26, 2020
trigonometry
Problem Statement
If
x
∈
R
\
{
k
π
2
∣
k
∈
Z
}
x \in \mathbb{R}\backslash \left \{\frac{k\pi}{2}\ |\ k\in \mathbb{Z} \right \}
x
∈
R
\
{
2
kπ
∣
k
∈
Z
}
, then
(
∑
0
≤
j
<
k
≤
n
sin
(
2
(
j
+
k
)
x
)
)
2
+
(
∑
0
≤
j
<
k
≤
n
cos
(
2
(
j
+
k
)
x
)
)
2
=
sin
2
n
x
sin
2
(
n
+
1
)
x
sin
2
x
sin
2
2
x
\left (\sum \limits_{0\leq j<k\leq n} \sin (2(j+k)x)\right )^2 + \left (\sum \limits_{0\leq j<k\leq n} \cos (2(j+k)x)\right )^2 = \frac{\sin ^2 nx \sin ^2 (n+1)x}{\sin ^2x \sin^22x}
0
≤
j
<
k
≤
n
∑
sin
(
2
(
j
+
k
)
x
)
2
+
0
≤
j
<
k
≤
n
∑
cos
(
2
(
j
+
k
)
x
)
2
=
sin
2
x
sin
2
2
x
sin
2
n
x
sin
2
(
n
+
1
)
x
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