Given:
2csc^2x-cot^4x=-1
cscx=1/sinx
cotx=cosx/sinx
2(1/sinx)^2-(cosx/sinx)^4=-1
2/sin^2x-cos^4x/sin^4x=-1
Multiplying throughout by sin^4x
2sin^2x-cos^4x=-sin^4x
Transposing
2sin^2x=cos^4x+sin^4x
cos^4x+sin^4x=(cos^2x+sin^2x)^2-2cos^2xsin^2x
2sin^2x=(cos^2x+sin^2x)^2-2cos^2xsin^2x
cos^2x+sin^2x=1
2sin^2x=(1)^2-2cos^2xsin^2x
2sin^2x=1-2cos^2xsin^2x
If
u=cos^2x
1-u=sin^2x
Substituting
2(1-u)=1-2u(1-u)
2-2u=1-2u+2u^2
2u^2-2u+2u+1-2=0
2u^2-1=0
2u^2=1
u^2=1/2
u=+-(1/sqrt2)
u=cos^2x
cos^2x=+-(1/sqrt2)
Considering only positive value for real numbers
cos^2x=1/sqrt2
cosx=+-(1/sqrtsqrt2)
theta=cos^-1(1/sqrtsqrt2), 2pi-cos^-1(1/sqrtsqrt2)