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Without using a computer but using pen and paper only, can anyone please help me calculate, simplify / evaluate the following?

$$ \frac{\tan(180^{\circ}/7)}{\tan(360^{\circ}/7)} - \frac{\tan(360^{\circ}/7)}{\tan(540^{\circ}/7)} - \frac{\tan(540^{\circ}/7)}{\tan(180^{\circ}/7)} $$

It evaluates to $-9$. This result is given for reference.

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  • $\begingroup$ I assume you mean $180^\circ$ etc.? $\endgroup$
    – mrf
    Feb 16, 2015 at 8:38
  • $\begingroup$ @mrf: BTW, is your logo bi-polar coordinates map? $\endgroup$
    – Narasimham
    Feb 16, 2015 at 9:24
  • $\begingroup$ @Narasimham No, it's an excerpt of a plot of some rational function (color coding the argument). $\endgroup$
    – mrf
    Feb 16, 2015 at 9:28
  • $\begingroup$ @mrf sir do you have any solution without using computer? $\endgroup$
    – Deddy
    Feb 16, 2015 at 14:34
  • $\begingroup$ It may have to do with like complex $ z^7 + 1 =0 $ $\endgroup$
    – Narasimham
    Feb 16, 2015 at 18:26

1 Answer 1

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EDIT1:

Among all arguments the highest common factor is $ (\pi/7) $ and $ \tan ( m \pi/7) $ can be expanded in terms of $ T_t =\tan (\pi/7), $ as m is an integer.

With obvious multiple argument notation:

$$ F(t) = \dfrac{T_t}{T_{2 t}} -\dfrac{T_{2t}}{T_{3 t}} - \dfrac{T_{3t}}{T_{ t}} $$

$$ T_t = t $$

$$ T_{2t}= \dfrac{2t}{1 - 2 t^2} $$

$$ T_{3t}= \dfrac{3 t- t^3 }{1 - 3 t^2} $$

and simplify it.

But it does simplify to the constant $ ( -9) $ you gave !

EDIT2: Honest, I used computer just to cross-verify that hand work cannot simplify fractions.

EDIT3:

$ F( \tan7 t) = 0 $. But using only paper/pencil such conclusion may not be so obvious to come to.

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  • $\begingroup$ So is there any solution without computer help? $\endgroup$
    – Deddy
    Feb 16, 2015 at 11:39
  • $\begingroup$ I suppose it is there, but cumbersome and tedious. The computer has necessity wise made many of us lazy. $\endgroup$
    – Narasimham
    Feb 17, 2015 at 6:00

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