sin(2x)=cos(3x)=sin(π/2-3x);
2x=π/2-3x, 5x=π/2, x=π/10; cot(5x)=1/tan(5x)=1/tan(π/2)=0.
More generally sin(2x)=cos(3x)=sin(π/2-3x+2πn), where n is an integer.
2x=π/2-3x+2πn, 5x=π(½+2n)=(π/2)(1+4n),
x=(π/10)(4n+1); when n=1, x=π/2; n=2, x=9π/10; n=3, x=13π/10; n=6, x=5π/2, etc.
sin(2x)=sin(π-2x), therefore:
π-2x=π/2-3x+2πn, x=2πn-π/2=(π/2)(4n-1); when n=1, x=3π/2; n=2, x=7π/2; n=3, x=11π/2, etc.
4n±1 is always an odd number and the tangent of odd multiples of π/2 (90°) is infinite.
cot(5x)=cot((π/2)(4n+1))=0;
cot(5x)=cot((5π/2)(4n-1))=0.