# Trigonometric linear in $\tan(180/7),\tan(360/7),\tan(540/7)$

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.

• I assume you mean $180^\circ$ etc.? – mrf Feb 16 '15 at 8:38
• @mrf: BTW, is your logo bi-polar coordinates map? – Narasimham Feb 16 '15 at 9:24
• @Narasimham No, it's an excerpt of a plot of some rational function (color coding the argument). – mrf Feb 16 '15 at 9:28
• @mrf sir do you have any solution without using computer? – Deddy Feb 16 '15 at 14:34
• It may have to do with like complex $z^7 + 1 =0$ – Narasimham Feb 16 '15 at 18:26

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.

• So is there any solution without computer help? – Deddy Feb 16 '15 at 11:39
• I suppose it is there, but cumbersome and tedious. The computer has necessity wise made many of us lazy. – Narasimham Feb 17 '15 at 6:00