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Mathematics
Problem
Let
n
n
n
be a positive integer and
Δ
r
=
∣
2
r
−
1
n
C
r
1
n
2
−
1
2
n
n
+
1
cos
2
n
2
cos
2
n
cos
2
(
n
+
1
)
∣
,
\Delta_r=\begin{vmatrix} 2r-1 & {}^n C_r & 1\\ n^2-1 & 2^n & n+1\\ \cos^2 n^2 & \cos^2 n & \cos^2(n+1) \end{vmatrix},
Δ
r
=
2
r
−
1
n
2
−
1
cos
2
n
2
n
C
r
2
n
cos
2
n
1
n
+
1
cos
2
(
n
+
1
)
,
prove that
Δ
r
=
∣
r
2
+
r
r
+
1
r
−
2
r
2
+
3
r
−
1
3
r
3
r
−
3
r
2
+
2
r
+
3
2
r
−
1
2
r
−
1
∣
.
\Delta_r=\begin{vmatrix} r^2+r & r+1 & r-2\\ r^2+3r-1 & 3r & 3r-3\\ r^2+2r+3 & 2r-1 & 2r-1 \end{vmatrix}.
Δ
r
=
r
2
+
r
r
2
+
3
r
−
1
r
2
+
2
r
+
3
r
+
1
3
r
2
r
−
1
r
−
2
3
r
−
3
2
r
−
1
.
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