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How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ without using a calculator.

Related question: how do we prove that $\cos(\pi/5)\cos(2\pi/5) = 0.25$, also without using a calculator

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A related question: How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?. Perhaps we have some other questions here, where $\cos\frac{\pi}5$ or $\cos\frac{2\pi}5$ is calculated, but I've only found this one. –  Martin Sleziak Apr 12 '12 at 11:44
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@MartinSleziak: I have an elementary answer for both questions (both parts of this question, and the related question as well, since my solution uses the polynomial $4x^2-2x-1=t^2-t-1$ for $t=2x$, which has roots $t=\frac{1\pm\sqrt5}{2}$. –  bgins Apr 12 '12 at 16:18
    
I am not sure if this is what Chandrasekar had in mind, because I don't follow his argument, but I have another post which is either the simplest solution yet, or else is a long-winded restatement of Chandrasekar's more elegant but potentially hard to follow solution (if correct, which I cannot judge). –  bgins Apr 12 '12 at 23:14
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Does anyone know if there's a geometric proof of this? Given that this can be expressed in terms of the vertices of a dodecagon it seems like there ought to be one somewhere... –  Steven Stadnicki Apr 12 '12 at 23:32
    
I gave a proof of a more general statement than your "related question" here: mathoverflow.net/questions/16583/a-trigonometry-problem –  user641 Apr 13 '12 at 10:46
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7 Answers

up vote 22 down vote accepted

The complex roots of $x^5-1$ are: $$ \begin{align} x_1&=1\\ x_2&=\cos\frac{2\pi}5+i\sin\frac{2\pi}5\\ x_3&=-\cos\frac{\pi}5+i\sin\frac{\pi}5\\ x_4&=-\cos\frac{\pi}5-i\sin\frac{\pi}5\\ x_5&=\cos\frac{2\pi}5-i\sin\frac{2\pi}5 \end{align} $$ using Vieta's formulas you get $$0=x_1+x_2+\dots+x_5=1+2\left(\cos\frac{2\pi}5-\cos\frac\pi5\right)=0,$$ which yields your first equation.


From now on, let $\varphi=\frac{\pi}5$ (for brevity).

We know that $\cos2\varphi-\cos\varphi+\frac12=(2\cos^2\varphi-1)-\cos\varphi+\frac12=2\cos^2\varphi-\cos\varphi-\frac12=0$, i.e. $$2\cos^2\varphi-\cos\varphi=\frac12.$$ Now from $\cos2\varphi=\cos\varphi-\frac12$ you get $$\cos\varphi\cos2\varphi=\cos^2\varphi-\frac{\cos\varphi}2=\frac{2\cos^2\varphi-\cos\varphi}2=\frac14.$$

(Or, as suggested in Chandrasekhar's answer, from $2\cos^2\varphi-\cos\varphi=\frac12$ you can find the value of $\cos\varphi$ by solving the quadratic equation and taking the positive root. Once you know $\cos\varphi$, you can compute $\cos2\varphi$ and many other things. If you try it this way, you can check your result e.g. here.)

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Dear Martin, There seems to be a typo in the real part of $x_4$. Regards, –  Matt E Apr 12 '12 at 10:27
    
Thanks @MattE, it should be corrected now. –  Martin Sleziak Apr 12 '12 at 10:34
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Note that: $\cos{2x} = \cos^{2}{x} - \sin^{2}{x} = 2\:\cos^{2}{x} - 1$. Therefore you have $\cos \frac{2\pi}{5} = 2\:\cos^{2}\frac{\pi}{5} - 1$

Now, \begin{align*} \cos\frac{\pi}{5} - \cos\frac{2\pi}{5} = \cos\frac{\pi}{5} - 2\: \cos^{2}\frac{\pi}{5}+1 \end{align*} This is a quadratic equation of the form $2 x^{2} - x -1 =0$ and solving this will give you the value of $\cos\frac{\pi}{5}$ from which you can find the above value which you need.

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Perhaps it should be mentioned that the quadratic equation has two solutions; but only one of them is positive, so there is no doubt which of the two solutions is $\cos\frac\pi5$. –  Martin Sleziak Apr 12 '12 at 10:38
    
Chandrasekhar: I think the quadratic equation should be $2x^{2}-x-\frac12=0$ or $4x^2-2x-1=0$. (If I understand correctly, you get the equation from $\cos\frac\pi5-\cos\frac{2\pi}5=\frac12$.) –  Martin Sleziak Apr 12 '12 at 10:51
    
@Martin: I have just calculated value of $\cos\frac{\pi}{5}$ using that equation. Why should it be $\frac{1}{2}$. That is what i have to prove. –  user9413 Apr 12 '12 at 11:04
    
It seems that I misunderstood your argumentation (I thought you are using $\cos\frac\pi5-\cos\frac{2\pi}5=\frac12$ to calculate $\cos\frac\pi5$ and then to solve the second part of question.) Nevertheless, the roots of $2x^2-x-1=0$ are $1$ and $-\frac12$, see wolframalpha. –  Martin Sleziak Apr 12 '12 at 11:10
    
@MartinSleziak: Yeah i know that. $(2x+1) \cdot (x-1)$ –  user9413 Apr 12 '12 at 11:11
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If $a=\cos\frac{\pi}5$ and $b=\cos\frac{2\pi}5=2a^2-1$ (by the double-angle identity for cosine), we want to show that $$ 0=(x-a)(x+b)=x^2-(a-b)x-ab=x^2-\tfrac12x-\tfrac14, $$ i.e., that $a$ and $-b$ are roots of $4x^2-2x-1$. Note also that $4x^2+2x-1=4(x+a)(x-b)$ has roots $-a$ and $b$. What is special about the numbers $\{a,b,-b,-a\}$? They are the $x$-coordinates (real parts) of the nonreal $10$th roots of unity, $$ x+iy=e^{\pm\pi i\cdot\frac{k}5} \qquad \text{for} \qquad k\not\equiv0\pmod5. $$ But these satisfy the equation $(x+iy)^5=(-1)^k=\pm1$. Taking the imaginary part, we have $$ \eqalign{ 0 &= \Im\left[(x+iy)^5\right] \\ &= \Im\left[x^5+5x^4(iy)+10x^3(iy)^2+10x^2(iy)^3+5x(iy)^4+(iy)^5\right] \\ &= \Im\left[iy\left(5x^4+10x^2(iy)^2+(iy)^4\right)\right] \\ &= y\left[5x^4-10x^2y^2+y^4\right] \\ \implies 0 &= \left[5x^4-10x^2(1-x^2)+(1-x^2)^2\right] \\ &= 16x^4-12x^2+1 \\ &= 16x^4-8x^2+1 ~-~4x^2 \\ &= \left( 4x^2-1 \right) - \left( 2x \right)^2 \\ &= \left( 4x^2+2x-1\right)\left( 4x^2-2x-1\right) \,. } $$ So far, we have shown this has roots $\{\pm a,\pm b\}$. It only remains to show that $\{-a,b\}$ are the roots of the first factor and that the desired pair $\{a,-b\}$ splits the second quadratic factor into linear terms (as we wrote at the outset).

Perhaps there is a clever way to infer the correct linear order of this set of roots by noticing that $a>\cos\frac\pi4>b$ so that $a^2>\frac12>b^2$ and combining this with our identity $b=2a^2-1$ above.

I propose in stead to use that $a>b$ but then to notice that our quadratic factors are in fact parabolas, with roots that are very easy to order on the $x$-axis. If we let $t=2x$ (which preserves order), then we have $$ t^2\pm t-1 = t\,(t\pm1)-1 $$ which both pass through $(0,-1)$ and alternately pass through $(\mp1,-1)$, shown respectively in red and blue below; the result follows.

ordered roots

It's only a small extra step to note that the roots of our factor, $t^2-t-1$, are $t=\frac{1\pm\sqrt5}{2}$, or $x=\frac{1\pm\sqrt5}{4}$.

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$$\cos(\pi/5) - \cos(2\pi/5)= {\rm Re}(e^{i\frac{\pi}{5}}-e^{i\frac{2\pi}{5}})=\frac{1}{2}( e^{i\frac{\pi}{5}}-e^{i\frac{2\pi}{5}}+ \overline{e^{i\frac{\pi}{5}}-e^{i\frac{2\pi}{5}}})=\frac{1}{2}( e^{i\frac{\pi}{5}}-e^{i\frac{2\pi}{5}}+ e^{-i\frac{\pi}{5}}-e^{-i\frac{2\pi}{5}})=\frac{e^{\frac{-2i\pi}{5}}}{2}( -e^{i\frac{4\pi}{5}}+e^{i\frac{3\pi}{5}}+ e^{i\frac{\pi}{5}}-1)$$

To simplify the computations, let $\omega=e^{\frac{i \pi}{5}}$. Note that $\omega^5=-1$.

Then

$$\cos(\pi/5) - \cos(2\pi/5)= \frac{-1}{2\omega^2}(\omega^4-\omega^3-\omega+1)= \frac{-1}{2\omega^2}(\omega^4-\omega^3+\omega^2-\omega+1-\omega^2)$$

$$\cos(\pi/5) - \cos(2\pi/5)= \frac{-1}{2\omega^2}(\frac{\omega^5+1}{\omega+1} -\omega^2)=\frac{-1}{2\omega^2}(0 -\omega^2)=\frac{1}{2}$$

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For the first equality.
$\cos(\pi/5) - \cos(2\pi/5)=\cos(3\pi/10-\pi/10) - \cos(3\pi/10+\pi/10)$
$=2\sin(\pi/10)\sin(3\pi/10)=2\sin(\pi/10)\cos(\pi/5)$
$=2\sin(\pi/10)\cos(\pi/10)\cos(\pi/5)/\cos(\pi/10)$
$=\sin(\pi/5)\cos(\pi/5)/\cos(\pi/10)$
$=0.5\sin(2\pi/5)/\cos(\pi/10)=0.5$

Here 's the latter equlity.
$\cos(\pi/5)\cos(2\pi/5) = \sin(\pi/5)\cos(\pi/5)\cos(2\pi/5)/\sin(\pi/5)$
$=0.5\times \sin(2\pi/5)\cos(2\pi/5)/\sin(\pi/5)=0.25\times \sin(4\pi/5)/\sin(\pi/5)$
$=0.25.$
Sorry for the bad composing. I am not familiar with tex.

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the most important one is \frac{a}{b}. I also like to use \cdot or no multiplication symbol at all. –  NeuroFuzzy Apr 13 '12 at 1:37
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I was not able (yet) to follow Chandrasekar's solution, but noticed this while trying to understand the argument (how it could possibly lead to the solution, or how exactly he arrives at $2x^2-x-1$ for $x=\cos\frac{\pi}{5}$, which to me seems non-obvious and even a fallacious deduction from his equations and prose -- apologies if I am just being dense)...perhaps it is what Chandrasekar meant all along, but in any case, it does seem to be the most elementary solution available.

Apply the double angle formula $\cos2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1$ to $\theta=\frac{\pi}{5}$ and $\frac{2\pi}{5}$, with $a=\cos\frac{\pi}{5}$ and $b=\frac{2\pi}{5}$ for convenience, recalling also that $\cos(\pi\pm\theta)=-\cos\theta$: $$ b=\cos\frac{2\pi}{5}=2\,\cos^2\frac{\pi}{5}-1=2a^2-1 $$ $$ -a=\cos\frac{4\pi}{5}=2\,\cos^2\frac{2\pi}{5}-1=2b^2-1 $$ Next, subtracting the equations $$ \matrix{ 2a^2=1+b\\ 2b^2=1-a} $$ we get

$$ \eqalign{ 2\left(a^2-b^2\right)&=b+a\\ 2\left(a+b\right)\left(a-b\right)&=b+a\\ 2\left(a-b\right)&=1\\ a-b&=\frac12\,. } $$ Furthermore, multiplying, we get $$ 4(ab)^2=(1+b)(1-a)=1+(b-a)-ab=1+\left(-\tfrac12\right)-ab $$ giving us the quadratic equation $$4(ab)^2+(ab)-\tfrac12=0$$ $$8(ab)^2+2(ab)-1=0$$ $$\left(4ab-1\right)\left(2ab+1\right)=0$$ so that $ab=\frac14$ or $-\frac12$, from which we can choose the former since we know that $0<a<b$.

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This is a very nice solution. –  Martin Sleziak Apr 13 '12 at 5:47
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$$\cos(\pi/5)\cos(2\pi/5)=A$$

$$\Longrightarrow\quad A = \frac{\sin(\pi/5)\cos(\pi/5)\cos(2\pi/5)}{\sin(\pi/5)}=\frac{\sin(2\pi/5)\cos(2\pi/5)}{2\sin(\pi/5)}$$

$$A = \frac{\sin(4\pi/5)} {2\cdot 2\cdot\sin(\pi/5)}=\frac{1}{4}$$ since $\sin(4\pi/5)=\sin(\pi/5)$.

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