# Funny identities [closed]

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?

## closed as primarily opinion-based by Najib Idrissi, user98602, user147263, apnorton, user2345215Jan 31 '15 at 18:16

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• This is related. – J. M. is a poor mathematician Nov 3 '10 at 22:35
• I noticed that your expression can be also written as $\sin(x - y) \sin(x + y) = (\cos y + \cos x) (\cos y - \cos x)$ – Quixotic Nov 4 '10 at 11:09
• 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 '13 at 0:31
• Almost an identity: $$\sqrt{123456790}\approx 11111.11111\,.$$ – Jakob Werner Jul 12 '14 at 18:47

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 $p\not\mid k$.
If we define $P$ as the infinite lower triangular matrix where $P_{i,j} = \binom{i}{j}$ (we can call it the Pascal Matrix), then $$P^k_{i,j} = \binom{i}{j}k^{i-j}$$
where $P^k_{i,j}$ is the element of $P^k$ in the position $i,j.$
$$\int_{-\infty}^{\infty}{\sin\left(x\right) \over x}\,{\rm d}x = \pi\int_{-1}^{1}\delta\left(k\right)\,{\rm d}k$$