Why not expand it then?

(1 +sinX)(tanX -cotX) +2cosX = 0

tanX -cotX +sinXtanX -sinXcotX +2cosX = 0

tanX -cotX +sinX[sinX/cosX] -sinX[cosX/sinX] +2cosX = 0

tanX -cotX +[sin^2(X)/cosX] -cosX +2cosX = 0

tanX -cotX +[(1 -cos^2(X)) /cosX] +cosX = 0

tanX -cotX +[1/cosX -cosX] +cosX = 0

tanX -cotX +1/cosX = 0

sinX/cosX -cosX/sinX +1/cosX = 0

Clear the fractions, multiply both sides by sinXcosX,

sin^2(X) -cos^2(X) +sinX = 0

sin^2(X) -[1 -sin^2(X)] +sinX = 0

2sin^2(X) +sinX -1 = 0

(2sinX -1)(sinX +1) = 0

2sinX -1 = 0

sinX = 1/2

X = 30deg, 150deg

sinX +1 = 0

sinX = -1

X = 270 deg

The 30 and 150 degrees check with the original equation, but the 270 degress did not.

Therefore, in the interval 0 <= X <= 360deg,

X = 30 deg or 150 deg -------------------------answer.