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1988 IMO Longlists
44
Deduce cosecant exression
Deduce cosecant exression
Source: IMO LongList 1988, Ireland 3, Problem 44 of ILL
November 3, 2005
trigonometry
algebra unsolved
algebra
Problem Statement
Let
−
1
<
x
<
1.
-1 < x < 1.
−
1
<
x
<
1.
Show that
∑
k
=
0
6
1
−
x
2
1
−
2
⋅
x
⋅
cos
(
2
⋅
π
⋅
k
7
)
+
x
2
=
7
⋅
(
1
+
x
7
)
(
1
−
x
7
)
.
\sum^{6}_{k=0} \frac{1 - x^2}{1 - 2 \cdot x \cdot \cos \left( \frac{2 \cdot \pi \cdot k }{7} \right) + x^2} = \frac{7 \cdot \left( 1 + x^7 \right)}{\left( 1 - x^7 \right)}.
k
=
0
∑
6
1
−
2
⋅
x
⋅
cos
(
7
2
⋅
π
⋅
k
)
+
x
2
1
−
x
2
=
(
1
−
x
7
)
7
⋅
(
1
+
x
7
)
.
Deduce that
csc
2
(
x
+
π
7
)
+
csc
2
(
2
⋅
x
+
π
7
)
+
csc
2
(
3
⋅
x
+
π
7
)
=
8.
\csc^2\left( x + \frac{\pi}{7} \right) + \csc^2\left(2 \cdot x + \frac{\pi}{7} \right) + \csc^2\left(3 \cdot x + \frac{\pi}{7} \right) = 8.
csc
2
(
x
+
7
π
)
+
csc
2
(
2
⋅
x
+
7
π
)
+
csc
2
(
3
⋅
x
+
7
π
)
=
8.
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