Trigonometry equation tan ⁡ 2 x + 3 sec ⁡ x + 3 = 0 for tan ⁡ 2 x + 3 sec ⁡ x + 3 = 0

Question

Trigonometry equation   tan  ⁡  2  x  +  3  sec  ⁡  x  +  3  =  0 for   tan  ⁡  2  x  +  3  sec  ⁡  x  +  3  =  0

Zackery Harvey · Accepted Answer

Clearly,   cos  ⁡  2  x  cos  ⁡  x  ≠  0Multiply both sides of the given equation  cos  ⁡  x  sin  ⁡  2  x  +  3  (  1  +  cos  ⁡  x  )  cos  ⁡  2  x  =  0  0  =  2  sin  ⁡  x      cos    2    ⁡  x  +  3  (  1  +  cos  ⁡  x  )  cos  ⁡  2  x  =  4  sin  ⁡            x      2        cos  ⁡            x      2            cos    2    ⁡  x  +  3  ⋅  2      cos    2    ⁡            x      2        cos  ⁡  2  x  =  2  cos  ⁡            x      2            (    2    sin    ⁡                  x        2                    cos      2        ⁡    x    +    3    cos    ⁡                  x        2              cos    ⁡    2    x    )  If   cos  ⁡            x      2        =  0  ,            x      2        =  (  2  n  +  1  )      90    ∘      ⟹    x  =  (  2  n  +  1  )      180    ∘    ≡      180    ∘      (  mod        360    ∘    )Else   2  sin  ⁡            x      2            cos    2    ⁡  x  +  3  cos  ⁡            x      2        cos  ⁡  2  x  =  0  −  2  tan  ⁡            x      2        =                    3        (        2                  cos          2                ⁡        x        −        1        )                              cos          2                ⁡        x            Use   cos  ⁡  x  =                    1        −                  tan          2                ⁡                              x            2                                      1        +                  tan          2                ⁡                              x            2                               which unfortunately leaves us with a bi-quadratic equation in   tan  ⁡            x      2

Trigonometry equation tan ⁡ 2 x + 3 sec ⁡

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