The characteristic property that three numbers ,  and   form an Arithmetic progression is this equality  
     -  =  - 



In our case, it means that

    3cos(x) - 2sin(x) = (sin(x) + 2cos(x)) - 3cos(x).


Simplify it

    3cos(x) - 2 cos(x) + 3cos(x) = sin(x) + 2sin(x)

    4cos(x)                      = 3sin(x).


Divide both sides by cos(x).  You will get

                     = ,   or

        tan(x)         = .


So, the first statement is proved.


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Next, if tan(x) =   and the angle x is acute,  then there is only one possibility:


    the angle x is in QI, and  sin(x) =  = 0.8,  cos(x) =  = 0.6.


In this case,  the 1st term of the AP is  2*0.8 = 1.6;  

               the 2nd term of the AP is  3*0.6 = 1.8  and

               the common difference is   1.8 - 1.6 = 0.2.


               Then the 20th term is 1.6+19*0.2 = 5.4  and

                    the sum of the first 20 terms is   = 70.