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)#
