Show that tan ⁡ 72 ∘ − cot ⁡ 36 ∘ − cot ⁡ 54 ∘ = 3 tan ⁡ 18 ∘

Question

Show that  tan  ⁡      72    ∘    −  cot  ⁡      36    ∘    −  cot  ⁡      54    ∘    =  3  tan  ⁡      18    ∘

billyfcash5n · Accepted Answer

A little generalization:Replacing       18    ∘   with t,            1              tan        ⁡        t              −  3  tan  ⁡  t  =                    4                  cos          2                ⁡        t        −        3                    sin        ⁡        t        cos        ⁡        t              =                    2        cos        ⁡        3        t                    cos        ⁡        t        sin        ⁡        2        t              =                    2        cos        ⁡        3        t        (        2        cos        ⁡        2        t        )                    cos        ⁡        t        sin        ⁡        4        t              =                    2        {        cos        ⁡        (        3        t        −        2        t        )        +        cos        ⁡        (        3        t        +        2        t        )        }                    cos        ⁡        t        sin        ⁡        4        t            if   cos  ⁡  t  ≠  0  and   cos  ⁡  5  t  =  0    ⟺    5  t  =  (  2  n  +  1  )      90    ∘      ⟺    t  =  (  2  n  +  1  )      18    ∘   where   n  ≢  2    (  mod    5  )Now   tan  ⁡  2  t  +            1              tan        ⁡        2        t              =            2              sin        ⁡        4        t            Here   n  =  0

uplakanimkk · Accepted Answer

cot  ⁡      36    ∘    +  cot  ⁡      54    ∘    =            2              sin        ⁡                  72          ∘                                  (  1  )  tan  ⁡      72    ∘    −  tan  ⁡      18    ∘    =                    sin        ⁡        (        72        −        18                  )          ∘                            cos        ⁡                  18          ∘                sin        ⁡                  18          ∘                      =                    2        sin        ⁡                  54          ∘                            sin        ⁡                  36          ∘                      =  2  cot  ⁡      36    ∘                (  2  )  cot  ⁡      36    ∘    −  tan  ⁡      18    ∘    =  cot  ⁡      36    ∘    −  cot  ⁡      72    ∘    =                    sin        ⁡        (        72        −        36                  )          ∘                            sin        ⁡                  36          ∘                sin        ⁡                  72          ∘                      =            1              sin        ⁡                  72          ∘                                  (  3  )    ⟹    tan  ⁡      72    ∘    −  cot  ⁡      36    ∘    −  cot  ⁡      54    ∘    −  3  tan  ⁡      18    ∘    =  (                    tan        ⁡                  72          ∘                −        tan        ⁡                  18          ∘                    ⏟        )  −  (                    cot        ⁡                  36          ∘                +        cot        ⁡                  54          ∘                    ⏟        )  −  2  tan  ⁡      18    ∘    =  2  (                    cot        ⁡                  36          ∘                −        tan        ⁡                  18          ∘                    ⏟        )  −            2              sin        ⁡                  72          ∘                      (  using   (  2  )  ,  (  1  )  )Now use (3)

Show that tan ⁡ 72 ∘ </msup> −<!-- - -

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