Find cos ⁡ 8 x if: tan ⁡ x tan ⁡ 2 x = cot ⁡ 2 x cot ⁡ 3 x

Question

Find   cos  ⁡  8  x if:  tan  ⁡  x  tan  ⁡  2  x  =  cot  ⁡  2  x  cot  ⁡  3  x

frethi38 · Accepted Answer

Let   t  :=  tan  ⁡  x and   T  :=      t    2    ≥  0 (assuming   x  ∈      R  ), so            4              T        2            (      3      −      T      )              (      1      −      T              )        2            (      1      −      3      T      )        =  1    ⟹    (  T  +  1  )  (      T    2    −  6  T  +  1  )  =  0    ⟹        T    2    −  6  T  +  1  =  0  ,and  cos  ⁡  4  x  =  2            (                        1          −          T                          1          +          T                    )        2    −  1  =                    T        2            −      6      T      +      1              (      1      +      T              )        2              =  0    ⟹    cos  ⁡  8  x  =  −  1.

Amber Quinn · Accepted Answer

Using the product-to-sum identity     tan  ⁡  θ  tan  ⁡  φ  =                    cos        ⁡        (        θ        −        φ        )        −        cos        ⁡        (        θ        +        φ        )                    cos        ⁡        (        θ        −        φ        )        +        cos        ⁡        (        θ        +        φ        )                                  (1)                    1        =                  tan          2                ⁡        2        x        ⋅                              cos            ⁡            2            x            −            cos            ⁡            4            x                                cos            ⁡            2            x            +            cos            ⁡            4            x                          =                              1            −                          cos              2                        ⁡            2            x                                              cos              2                        ⁡            2            x                          ⋅                              cos            ⁡            2            x            −            2                          cos              2                        ⁡            2            x            +            1                                cos            ⁡            2            x            +            2                          cos              2                        ⁡            2            x            −            1                              With     t  =  cos  ⁡  2  x                                t          2                (        2                  t          2                +        t        −        1        )        =        (        1        −                  t          2                )        (        −        2                  t          2                +        t        +        1        )                                                    ⟺                                2                  t          3                +        2                  t          2                −        t        −        1        =        0                                                  ⟺                                (        t        +        1        )        (        2                  t          2                −        1        )        =        0                                                  ⟺                                t        ∈                  {          −          1          ,          ±                                    2                        2                    }                    The root     t  =  −  1   must be excluded because the denominator in (1)  vanishes when     cos  ⁡  2  x  =  −  1  , which leaves     cos  ⁡  2  x  =  ±            2        2

Find cos ⁡ 8 x if: tan ⁡ x tan ⁡<!--

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