Introduce new variable

    y = sin(x) + .


Then y^2 = sin^2(x) + 2 + 1/sin^2(x),  and the original equation takes the form

    y^2 - 2 + y = 4,

or

    y^2 + y - 6 = 0.


Its roots are (using the quadratic formula)

     =  =  = ,

    or   =  = 2,   =  = -3.


It means that  sin(x) +   is  EITHER  2  OR  -3.


So, from this point, we have two cases for x.



(1)  sin(x) +  = 2,  which implies

     sin^2(x) - 2sin(x) + 1 = 0,

     (sin(x) - 1)^2 = 0

      sin(x) = 1

          x  = 90 degrees (or  x = ).

      Thus this case is complete.



(2)  sin(x) +  = -3,  which implies

     sin^2(x) + 3*sin(x) + 1 = 0,

     sin(x) = use the quadratic formula =  = . 

      =  = ~ -0.382  --->  from it, you may find two values for x between 0 and 360 degrees.

      =  = ~ -2.618  --->  there is no solution for real x. 

      Thus this case is complete, too.