Trying to solve the trig equation 3 + 4 cos 2 ⁡ ( x ) = sin ⁡ ( x ) 3 + 3 cos ⁡ ( x )

Question

Trying to solve the trig equation       3    +    4          cos      2        ⁡    (    x    )    =            sin      ⁡      (      x      )              3        +  3  cos  ⁡  (  x  )

Leslie Rollins · Accepted Answer

My solution goes like this      {                            3          +          4                      cos            2                    ⁡          (          x          )          =                                                    sin                2                            ⁡              (              x              )                        3                    +                      6                          3                                sin          ⁡          (          x          )          cos          ⁡          (          x          )          +          9                      cos            2                    ⁡          (          x          )                                                                sin              ⁡              (              x              )                                      3                                +          3          cos          ⁡          (          x          )          ≥          0                          3  (      sin    2    ⁡  (  x  )  +      cos    2    ⁡  (  x  )  )  +  4      cos    2    ⁡  (  x  )  =                    sin        2            ⁡      (      x      )        3    +      6          3        sin  ⁡  (  x  )  cos  ⁡  (  x  )  +  9      cos    2    ⁡  (  x  )  2      cos    2    ⁡  (  x  )  +                    sin        2            ⁡      (      x      )        3    −  3      sin    2    ⁡  (  x  )  +      6          3        sin  ⁡  (  x  )  cos  ⁡  (  x  )  =  0I multiply by 3 and divide by       cos    2    ⁡  (  x  ):  8      tan    2    ⁡  (  x  )  −  6      3    tan  ⁡  (  x  )  −  6  =  0Let   t  =  tan  ⁡  (  x  ), then  4      t    2    −  3      3    t  −  3  =  0    ⟹    t  =                    3                  3                ±                  (          3                      3                                )            2                    −          4          ⋅          4          (          −          3          )                            2        ⋅        4              =                    3                  3                ±        5                  3                    8        =  ?

Trying to solve the trig equation <msqrt> 3 + 4 cos 2 </

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