9tan(2x) − 18cos(x) = 0

Divide through by 9

                 tan(2x) − 2cos(x) = 0

                     - 2cos(x) = 0

                 - 2cos(x) = 0

Divide through by 2

                  - cos(x) = 0

Multiply through by LCD of 1 - 2sin²(x)

sin(x)cos(x) - cos(x)(1 - 2sin²(x)) = 0

Factor out cos(x)

     cos(x)[sin(x) - (1 - 2sin²(x)] = 0

      cos(x)[sin(x) - 1 + 2sin²(x)} = 0

Use the zero-factor property:

cos(x) = 0;            sin(x) - 1 + 2sin²(x) = 0

x = , ;            2sin²(x) + sin(x) - 1 = 0     

                   [2sin(x) - 1][sin(x) + 1] = 0

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

There are 4 solutions in  [0, 2π).

               , , , and                              

Edwin