Note that this is not an identity. It's an equation which has a specific solution.
cosec(2A)=1/sin(2A)=1/(2sinAcosA).
secA+cotA=1/cosA+cosA/sinA=(sinA+cos2A)/(sinAcosA).
Therefore:
1/(2sinAcosA)=(sinA+cos2A)/(sinAcosA),
½=sinA+1-sin2A, sin2A-sinA=½, complete the square:
sin2A-sinA+¼=¾,
(sinA-½)2=¾,
sinA-½=±√3/2, sinA=½-√3/2, A=338.53° approx.