# Funny identities

Here is a funny exercise $$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$ (If you prove it don't publish it here please). Do you have similar examples?

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Maybe a moderator should put the zeta ones together since there are three already? –  anon Nov 3 '10 at 22:29
Perhaps this should be a community wiki question. –  Nuno Nov 3 '10 at 22:31
This is related. –  Ｊ. Ｍ. Nov 3 '10 at 22:35
I have tripped up many calculus students with this one: $log(1+2+3)=log1+log2+log3$. I am evil... –  user641 Dec 8 '12 at 1:23
@SteveD If only we could find an odd example... –  peoplepower Jan 13 at 0:31

$$\int_0^1\frac{\mathrm{d}x}{x^x}=\sum_{k=1}^\infty \frac1{k^k}$$

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I had to do something about my accept range :) –  AD. May 17 '12 at 4:47
Sophomore's Dream? –  rotskoff Jun 19 '12 at 20:50
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$$\frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\frac{\eta(2k)}{2^{2k}}$$ $$\frac{\pi}{3}=1+2\sum_{k=1}^{\infty}\frac{\eta(2k)}{6^{2k}}$$ where $\eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}}$

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For all $n\in\mathbb{N}$ and $n\neq1$ $$\prod_{k=1}^{n-1}2\sin\frac{k \pi}{n} = n$$

For some reason, the proof involves complex numbers and polynomials.

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Best near miss

$$\int_{0}^{\infty }\cos\left ( 2x \right )\prod_{n=0}^{\infty}\cos\left ( \frac{x}{n} \right )~\mathrm dx\approx \frac{\pi}{8}-7.41\times 10^{-43}$$

One can easily be fooled into thinking that it is exactly $\dfrac{\pi}{8}$.

References:

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Let $\sigma(n)$ denote the sum of the divisors of $n$.

If $$p=1+\sigma(k),$$ then $$p^a=1+\sigma(kp^{a-1})$$ where $a,k$ are positive integers and $p$ is a prime such that $p\not\mid k$.

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$$\frac{1}{2}=\frac{\frac{1}{2}}{\frac{1}{2}+\frac{\frac{1}{2}}{\frac{1}{2}+\frac{\frac{1}{2}}{\frac{1}{2}+\frac{\frac{1}{2}}{\frac{1}{2}+\frac{\frac{1}{2}}{\frac{1}{2}+\frac{\frac{1}{2}}{\frac{1}{2}+\cdots}}}}}}$$

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$$27\cdot56=2\cdot756,$$ $$277\cdot756=27\cdot7756,$$ $$2777\cdot7756=277\cdot77756,$$ and so on.

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\begin{align}\frac{64}{16}&=\frac{6\!\!/\,4}{16\!\!/}\\&=\frac41\\&=4\end{align}

For more examples of these weird fractions, see "How Weird Are Weird Fractions?", Ryan Stuffelbeam, The College Mathematics Journal, Vol. 44, No. 3 (May 2013), pp. 202-209.

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$$\sin \theta \cdot \sin \bigl(60^\circ - \theta \bigr) \cdot \sin \bigl(60^\circ + \theta \bigr) = \frac{1}{4} \sin 3\theta$$

$$\cos \theta \cdot \cos \bigl(60^\circ - \theta \bigr) \cdot \cos \bigl(60^\circ + \theta \bigr) = \frac{1}{4} \cos 3\theta$$

$$\tan \theta \cdot \tan \bigl(60^\circ - \theta \bigr) \cdot \tan \bigl(60^\circ + \theta \bigr) = \tan 3\theta$$

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I just wanted to mention that your first identity is equivalent to the case $n=3$ of the formula for $\sin nx$ given there. (Just replace $\sin(60^{\circ}-\theta)$ by $\sin(\theta+120^{\circ})$.) –  Hans Lundmark Nov 4 '10 at 9:56
considering your first two identities the thirth should be $$\tan \theta \cdot \tan \bigl(60 - \theta \bigr) \cdot \tan \bigl(60 + \theta \bigr) = \tan 3\theta$$ –  Neves Mar 6 '11 at 16:08

$\textbf{Claim:}\quad$$\frac{\sin x}{n}=6$$ for all$n,x$($n\neq 0$).$\textit{Proof:}\quad$$\frac{\sin x}{n}=\frac{\dfrac{1}{n}\cdot\sin x}{\dfrac{1}{n}\cdot n}=\frac{\operatorname{si}x}{1}=\text{six}.\quad\blacksquare$$

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$\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3) = \pi$ (using the principal value), but if you blindly use the addition formula $\tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\dfrac{x+y}{1-x y}$ twice, you get zero:

$\tan^{-1}(1) + \tan^{-1}(2) = \tan^{-1}\dfrac{1+2}{1-1*2} =\tan^{-1}(-3)$; $\tan^{-1}(1) + \tan^{-1}(2) + \tan^{-1}(3) =\tan^{-1}(-3) + \tan^{-1}(3) =\tan^{-1}\dfrac{-3+3}{1-(-3)(3)} = 0$.

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$$\lim_{\omega\to\infty}3=8$$ The "proof" is by rotation through $\pi/2$. More of a joke than an identity, I suppose.

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$$2592=2^59^2$$ Found this in one of Dudeney's puzzle books

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$3435=3^3+4^4+3^3+5^5%$
\begin{array}{rcrcl} \vdots & \vdots & \vdots & \vdots & \vdots \1mm] \int{1 \over x^{3}}\,{\rm d}x & = & -\,{1 \over 2}\,{1 \over x^{2}} & \sim & x^{\color{#ff0000}{\large\bf -2}} \\[1mm] \int{1 \over x^{2}}\,{\rm d}x & = & -\,{1 \over x} & \sim & x^{\color{#ff0000}{\large\bf -1}} \\[1mm] \int{1 \over x}\,{\rm d}x & = & \ln\left(x\right) & \sim & x^{\color{#0000ff}{\LARGE\bf 0}} \color{#0000ff}{\LARGE\quad ?} \\[1mm] \int x^{0}\,{\rm d}x & = & x^{1} & \sim & x^{\color{#ff0000}{\large\bf 1}} \\[1mm] \int x\,{\rm d}x & = & {1 \over 2}\,x^{2} & \sim & x^{\color{#ff0000}{\large\bf 2}} \\[1mm] \vdots & \vdots & \vdots & \vdots & \vdots \end{array} - show 3 more comments \int_{-\infty}^{\infty}{\sin\left(x\right) \over x}\,{\rm d}x = \pi\int_{-1}^{1}\delta\left(k\right)\,{\rm d}k - add comment \sqrt{\vphantom{\large A}2025\,} = 20 + 25 = 45 - add comment \begin{align} E &= \sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} = mc^{2} + \left[\sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} - mc^{2}\right] \\[3mm]&= mc^{2} + {\left(pc\right)^{2} \over \sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} + mc^{2}} = mc^{2} + {p^{2}/2m \over 1 + {\sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} - mc^{2} \over 2mc^{2}}} \\[3mm]&= mc^{2} + {p^{2}/2m \over 1 + {p^{2}/2m \over \sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} + mc^{2}}} = mc^{2} + {p^{2}/2m \over 1 + {p^{2}/2m \over 1 + {p^{2}/2m \over \sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} - mc^{2}}}} \end{align} - add comment Amazing one : \begin{align} \exp(\pi \text{i}) = -1 \end{align} - add comment \[\sqrt{n^{\log n}}=n^{\log \sqrt{n}} - a^{\log b} = b^{\log a} for a and b at least 1. – Wok Nov 30 '10 at 10:03 You should have written \sqrt{n^{\log n}} – ypercube Feb 25 '11 at 22:13 show 1 more comment \begin{align} \frac{d}{dx}(x^x) &= x\cdot x^{x-1} &\text{Power Rule?}&\text{False}\\ \frac{d}{dx}(x^x) &= x^{x}\ln(x) &\text{Exponential Rule?}&\text{False}\\ \frac{d}{dx}(x^x) &= x\cdot x^{x-1}+x^{x}\ln(x) &\text{Sum of these?}&\text{True}\\ \end{align} - add comment Here is an Asian kid paradox - perhaps they will understand (apologies for not being strictly mathematical). IfStudy=No \;Fail$$and$$No \; Study=Fail$$then$$Study+No\;Study=Fail+No\;Fail\implies (1+No)Study=(1+No)Fail$$Cancelling gives$$Study=Fail$$Isn't that weird??? - 1+No is 0 in both sides. You can't cancel zero... – CODE Jun 4 at 16:47 add comment I have one: In a \Delta ABC,$$\tan A+\tan B+\tan C=\tan A\tan B\tan C.$$- Also \cot(A/2)+\cot(B/2)+ \cot(C/2)=\cot(A/2)\cot(B/2)\cot(C/2). – N. S. Apr 11 at 22:39 add comment We have by block partition rule for determinant$$ \det \left[ \begin{array}{cc} U & R \\ L & D \end{array} \right] = \det U\cdot \det ( D-LU^{-1}R) $$But if U,R,L and D commute we have that$$ \det \left[ \begin{array}{cc} U & R \\ L & D \end{array} \right] = \det (UD-LR) $$- add comment The following number is prime p = 785963102379428822376694789446897396207498568951 and p in base 16 is 89ABCDEF012345672718281831415926141424F7 which includes counting in hexadecimal, and digits of e, \pi, and \sqrt{2}. Do you think this's surprising or not?$$11 \times 11 = 121111 \times 111 = 123211111 \times 1111 = 123432111111 \times 11111 = 123454321\vdots$$- show 2 more comments 32768=(3-2+7)^6 / 8 Just a funny coincidence. - add comment$$ 10^2+11^2+12^2=13^2+14^2 $$There's a funny Abstruse Goose comic about this, which I can't seem to find at the moment. - abstrusegoose.com/63 – Memming Sep 27 at 22:59 add comment Well, i don't know whether to classify this as funny or surprising, but ok it's worth posting. • Let (X,\tau) be a topological space and let A \subset X . By iteratively applying operations of closure and complemention, one can produce at most 14 distinct sets. It's called as the Kuratowski's Closure complement problem. - An example achieving the is [0,1] \cup (2,3) \cup \{(4,5) \cap \mathbb{Q}\} \cup \{(6,8) - \{7\}\} \cup \{9\}. See section 9 of austinmohr.com/Work_files/730.pdf for details. – Austin Mohr Jun 19 '12 at 2:09 I think you mean [0, 1]\cup (2, 3)\cup((4, 5)\cap\mathbb{Q})\cup(6, 7)\cup(7, 8)\cup\{9\}. The set you wrote isn't a subset of \mathbb{R} as it contains (4, 5)\cap\mathbb{Q} as an element. – Michael Albanese Jan 13 at 15:08 add comment By excluding the first two primes, Euler's Prime Product becomes a square:$$\prod _{n=3}^{\infty } \frac{1}{1-\frac{1}{(p_n)^{2}}}=\frac{\pi ^2}{9}$$By using multiples of the product of the first two primes, we get the square root:$$\prod _{n=1}^{\infty } \frac{1}{1-\frac{1}{(n p_1 p_2)^{2}}}=\frac{\pi }{3}$$- It doesn't make sense to speak of "perfect squares" for positive real numbers... but this is a nice identity though. – Patrick Da Silva Jun 19 '12 at 19:53 @PatrickDaSilva It might, if you know that the values of L-functions sometimes land in a special ring which is strictly between algebraic numbers and transcendental numbers. This is the ring of 'periods'. I don't believe that it is closed under taking square roots, so to say that something is the square of a period might not be completely silly. – Bruno Joyal Sep 26 at 21:29 add comment$$\frac{1}{998901}=0.000001002003004005006...997999000001...