Trigonometry with multiple angle and exact value of $\tan\pi/5$

By considering the equation $\tan5\theta=0$, show that the exact value of $\tan\pi/5$ is $\sqrt{5-2\sqrt{5}}$.

Do I need to evaluate the multiple angle for $\tan5\theta=0$?

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There is only one $\theta$ strictly between $0$ and $\frac{\pi}{2}$ such that $\tan 5\theta=0$. –  André Nicolas Feb 4 '13 at 22:14
I do it for $e^{i \pi / 5}$ which gives the cosine and the sine. –  Will Jagy Feb 4 '13 at 22:15
See also the discussions here: math.stackexchange.com/questions/275151/… –  Ron Gordon Feb 4 '13 at 22:37

$$\tan(5x) = \dfrac{\tan(3x) + \tan(2x)}{1-\tan(3x) \tan(2x)} = \dfrac{\dfrac{3 \tan(x) - \tan^3(x)}{1-3 \tan^2(x)} + \dfrac{2 \tan(x)}{1-\tan^2(x)}}{1- \dfrac{3 \tan(x) - \tan^3(x)}{1-3 \tan^2(x)} \cdot \dfrac{2 \tan(x)}{1-\tan^2(x)}}$$ Hence, $$\tan(5x) = 0 \implies (3t-t^3)(1-t^2) + 2t(1-3t^2) = t \left(t^4-4t^2+3 - 6t^2+2 \right) = 0$$ where $t= \tan(x)$. Since we are interested in $\tan(\pi/5)$, we can rule out $t=0$. Hence, we need to solve a bi-quadratic $t^4-10t^2+5 = 0$. This gives us $$t^2 = \dfrac{10 \pm \sqrt{100-4 \times 5}}{2} = 5 \pm 2 \sqrt5 \implies t = \pm \sqrt{5 \pm 2\sqrt5}$$ We also know that $\tan(\pi/5) \in (\tan(0),\tan(\pi/4)) = \left(0,1 \right)$. Hence, $$\tan(\pi/5) = \sqrt{5 - 2\sqrt5}$$

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I don't understand $\in$ notation and the stuff following it. –  bbr4in Feb 4 '13 at 22:40
@user52187 Since $\tan$ is an increasing function, we have $\tan(0) < \tan(\pi/5) < \tan(\pi/4)$. Hence, $0 < \tan(\pi/5) < 1$. Of the four values $\pm \sqrt{5 \pm 2\sqrt5}$, only $\sqrt{5 - 2\sqrt5}$ lies in the interval $(0,1)$. –  user17762 Feb 4 '13 at 22:43

The idea is to expand $\tan 5\theta$ so as to get a function of $\tan \theta$. Then by letting $\theta = \pi/5$, you will get an equation in $\tan \pi/5$.

In details, put $x = \tan \pi/5$ $$\tan 5 \theta = \frac{\tan 2\theta + \tan 3\theta}{1 - \tan 2\theta \tan 3\theta}$$

Then you find that $\tan 2\pi/5 + \tan 3\pi/5 = 0$.

But $$\tan 2\pi/5 = \frac{2 x}{1 - x^2}$$

And $$\tan 3\pi/5 = \frac{\tan(2\pi/5) + x}{1 - x\tan(2\pi/5)} = \frac{3x - x^3}{1 - 3x^2}.$$

Conclusion $$\frac{2x}{1 -x^2} + \frac{3x - x^3}{1 - 3x^2} = 0.$$

Because $x \neq 0$, we see that $x$ is solution of the polynomial $x^4 - 10x^2 +5 = 0$.

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Using Euler's Formula and Binomial Theorem, we get \begin{align} \cos(5\theta)+i\sin(5\theta) &=\left(\cos(\theta)+i\sin(\theta)\right)^5\\ &=\cos^5(\theta)+5i\cos^4(\theta)\sin(\theta)-10\cos^3(\theta)\sin^2(\theta)\\ &-10i\cos^2(\theta)\sin^3(\theta)+5\cos(\theta)\sin^4(\theta)+i\sin^5(\theta)\tag{1} \end{align} Taking the ratio of the real and imaginary parts of $(1)$, we get $$\tan(5\theta)=\frac{5\tan(\theta)-10\tan^3(\theta)+\tan^5(\theta)}{1-10\tan^2(\theta)+5\tan^4(\theta)}\tag{2}$$ Thus, if $\tan(5\theta)=0$, but $\tan(\theta)\ne0$, $(2)$ says that $$5-10\tan^2(\theta)+\tan^4(\theta)=0\tag{3}$$ The Quadratic Formula yields $$\tan^2(\theta)=5\pm2\sqrt{5}\tag{4}$$ Therefore, $$\tan(\theta)=\pm\sqrt{5\pm2\sqrt{5}}\tag{5}$$ Matching up the least positive values of $\theta$ for which $\tan(5\theta)=0$ yields \begin{align} \tan\left(\frac\pi5\right)&=+\sqrt{5-2\sqrt{5}}\\ \tan\left(\frac{2\pi}5\right)&=+\sqrt{5+2\sqrt{5}}\\ \tan\left(\frac{3\pi}5\right)&=-\sqrt{5+2\sqrt{5}}\\ \tan\left(\frac{4\pi}5\right)&=-\sqrt{5-2\sqrt{5}}\\ \end{align}\tag{6} Note that these values support this result.

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