cot(5X)=-1, 1/tan(5X)=-1, tan(5X)=-1, 5X=3π/4 (quadrant 2), 7π/4 (quadrant 4). The quadrants are separated by π radians. 5X=(4n+3)π/4, where n is an integer.
There are 10 solutions (4n+3)π/20 in various quadrants (Q1-Q4):
n X
0 3π/20 Q1
1 7π/20 Q1
2 11π/20 Q2
3 3π/4 Q2
4 19π/20 Q2
5 23π/20 Q3
6 27π/20 Q3
7 31π/20 Q4
8 7π/4 Q4
9 39π/20 Q4