Calculate tan 2 ⁡ π 5 + tan 2 ⁡ 2 π 5 without a calculatorThe question is to find the exact value of: tan 2 ⁡ ( π 5 ) + tan 2 ⁡ ( 2 π 5 ) without using a calculator.I know that it is possible to find the exact values of tan ⁡ ( π 5 ) and tan ⁡ ( 2 π 5 ) to find that the answer is 10. However, I want to know whether there is a faster way that does not involve calculating those values.So far, I have this: let a = tan ⁡ ( π 5 ) and b = tan ⁡ ( 2 π 5 ) ; then, b = 2 a 1 − a 2 and a = − 2 b 1 − b 2 , so multiplying the two and simplifying gives: a 2 + b 2 = ( a b ) 2 + 5Any ideas? Thanks!

Question

Calculate       tan    2    ⁡            π      5        +      tan    2    ⁡                    2        π            5       without a calculatorThe question is to find the exact value of:      tan    2    ⁡            (              π        5            )        +      tan    2    ⁡            (                        2          π                5            )      without using a calculator.I know that it is possible to find the exact values of   tan  ⁡            (              π        5            )       and   tan  ⁡            (                        2          π                5            )       to find that the answer is   10. However, I want to know whether there is a faster way that does not involve calculating those values.So far, I have this: let   a  =  tan  ⁡            (              π        5            )       and   b  =  tan  ⁡            (                        2          π                5            )      ; then,   b  =            2      a              1      −              a        2             and   a  =  −            2      b              1      −              b        2            , so multiplying the two and simplifying gives:      a    2    +      b    2    =                    (        a        b        )              2    +  5Any ideas? Thanks!

toroipanfz · Accepted Answer

Let   a  =  tan  ⁡  (      π    5    ) and   b  =  tan  ⁡  (            2      π        5    ). Also let   s  =      a    2    +      b    2   and   p  =  (  a  b      )    2  . We have                              a          2                +                  b          2                                    =                    (        a        b                  )          2                +        5                            s                            =                    p        +        5.            Now   a  (  1  −      b    2    )  =  −  2  b and   b  (  1  −      a    2    )  =  2  a square these equations and add them together                    3        (                  a          2                +                  b          2                )        +        4        (        a        b                  )          2                                    =                    (        a        b                  )          2                (                  a          2                +                  b          2                )                            3        s        +        4        p                            =                    s        p        .            Now eliminate   p and we have the quadratic       s    2    −  12  s  −  20  =  0. This has roots   2 and       10

Desirae Arias · Accepted Answer

tan    2    ⁡      π    5    +      tan    2    ⁡            2      π        5    =            1      −      cos      ⁡                        2          π                5                    1      +      cos      ⁡                        2          π                5              +            1      −      cos      ⁡                        4          π                5                    1      +      cos      ⁡                        4          π                5              =  =            2      −      2      cos      ⁡                        2          π                5            cos      ⁡                        4          π                5                    1      +      cos      ⁡                        2          π                5            +      cos      ⁡                        4          π                5            +      cos      ⁡                        2          π                5            cos      ⁡                        4          π                5              =  =            2      −                        4          sin          ⁡                                    2              π                        5                    cos          ⁡                                    2              π                        5                    cos          ⁡                                    4              π                        5                                    2          sin          ⁡                                    2              π                        5                                      1      +                        2          sin          ⁡                      π            5                    cos          ⁡                                    2              π                        5                    +          2          sin          ⁡                      π            5                    cos          ⁡                                    4              π                        5                                    2          sin          ⁡                      π            5                              +                        4          sin          ⁡                                    2              π                        5                    cos          ⁡                                    2              π                        5                    cos          ⁡                                    4              π                        5                                    4          sin          ⁡                                    2              π                        5                                =  =            2      −                        sin          ⁡                                    8              π                        5                                    2          sin          ⁡                                    2              π                        5                                      1      +                        sin          ⁡                                    3              π                        5                    −          sin          ⁡                      π            5                    +          sin          ⁡                                    5              π                        5                    −          sin          ⁡                                    3              π                        5                                    2          sin          ⁡                      π            5                              +                        sin          ⁡                                    8              π                        5                                    4          sin          ⁡                                    2              π                        5                                =            2      +              1        2                    1      −              1        2            −              1        4              =  10.

Calculate tan^2 pi/5+tan^2 (2pi)/5 without a calculator I know that it is possible to find the exact values of tan(pi/5) and tan((2pi)/5) to find that the answer is 10. However, I want to know whether there is a faster way that does not involve calculating those values.

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