find all solutions 2cotxcosx-3cosx+6cotx-9=0

Question

Find all solutions between 0 and 2pi; 2cotxcosx-3cosx+6cotx-9=0

Answer 1

2 CotX cos X - 3 CosX + 6 CotX - 9 =0

 

  2 Cos ²X - 3 Cos X + 6 Cos X - 9 =0

     sin x                        sin x

 

   2 Cos² X  - 3 CosX Sin X + 6 CosX - 9SinX =0

                Sin X

 

  2 Cos²X + 6 CosX - 3CosX - 9sinX = 0

 

   2 CosX ( cos X + 3 ) - 3 ( CosX + 3 sin X) =0

 

    (  Cos x + 3 ) ( 2CosX -3 SinX) = 0

 

      Cos X + 3 = 0   Cos X = -3  ( No Solution)

 

    2 COs X - 3SinX = 0     tanx = 2/3  

 

     X = 33.69° , 213.69°

Answer 2

2 CotX cos X - 3 CosX + 6 CotX - 9 =0

(2 Cos2 X - 3 CosX Sin X + 6 CosX - 9SinX)/sinx =0

Then, multiplying the two sides by sin x we have

2 Cos2X + 6 CosX - 3CosX sinX  - 9sinX = 0

after factorization,

2 CosX ( cos X + 3 ) - 3sinX ( CosX + 3 ) =0

factorizing again we obtain

(cosX + 3) ( 2 cosX - 3 sinX)=0

First answer making (cosX + 3)=0

Cos X = 3 not possible

Second: (2cosX - 3 sinX) = 0 => 2 - 3 tgX =0 => tgX = (2/3)

x = 33.69 degrees

CHECK!!!!!!!!!!!!!!!!!!