Famous or common mathematical identities that yield $1$

Question

To me the most common identity that comes to mind that results in $1$ is the trigonometric sum of squared cosine and sine of an angle:

$$ \cos^2{\theta} + \sin^2{}\theta = 1 \tag{1} $$ and maybe

$$ -e^{i\pi} = 1 \tag{2} $$

Are there other famous (as in commonly used) identities that yield $1$ in particular?

Answer 1

Here are two well-known infinite series whose sum is $1$: $$\sum_{n=1}^{\infty} \frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots = 1$$ $$\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + \frac{1}{5 \cdot 6} + \cdots = 1$$ The following series is less known, interesting although: $$\sum_{n=1}^{\infty} \frac{1}{s_n} = \frac{1}{2} + \frac{1}{3} + \frac{1}{7} + \frac{1}{43} + \frac{1}{1807} + \cdots = 1$$ where denominators form Sylvester sequence: every term is equal to the product of all previous terms, plus one. For example $$\begin{array}{rcl} 2& & \\ 3 & = & 2+1\\ 7 & = & 2 \cdot 3 +1 \\ 43 & = & 2 \cdot 3 \cdot 7 +1 \\ 1807 & = & 2 \cdot 3 \cdot 7 \cdot 43 +1 \\ \end{array}$$ and so on.

Answer 2

Pick your favorite probability density function $f$, then $$ \int_{-\infty}^\infty f(x)dx=1. $$

Answer 3

$$(\forall x\in \Bbb R)\;\;\cosh^2(x)-\sinh^2(x)=1$$

Answer 4

$$\sec^2x-\tan^2x = 1$$ $$\csc^2x-\cot^2x = 1$$ $$\sin x \csc x = 1$$ $$\tan x\cot x = 1$$ $$\sec x\cos x = 1$$ $$ \frac{\sin x + \cos x}{\cos ^3x} - \tan^3x - \tan^2x-\tan x = 1$$

Answer 5

Limits $$\lim_{x\to0}\frac{\sin x}{x}=1$$ This is actually a special case of the more general identity: If $f(0)=0$ and $f'(0)=1$, then $$\lim_{x\to0}\frac{f(x)}{x}=1$$ which includes things like $$\lim_{x\to0}\frac{\arctan x}{x}=1$$

Series

Any geometric series (which converges) will do as long as the first term, $a$, is equal to $1-r$, where $r$ is the common ratio. For example: $$\begin{align}1&=\frac{1}{3}+\frac{1}{3}\left(\frac{2}{3}\right)+\frac{1}{3}\left(\frac{2}{3}\right)^2+\frac{1}{3}\left(\frac{2}{3}\right)^3+\cdots\\ &=\frac{1}{4}+\frac{1}{4}\left(\frac{3}{4}\right)+\frac{1}{4}\left(\frac{3}{4}\right)^2+\frac{1}{4}\left(\frac{3}{4}\right)^3+\cdots\\ &=\sum_{k=1}^\infty (1-r)r^{k-1},~~~~~~~\lvert r\rvert <1\end{align}$$ Another quite interesting series: $$\sum_{k=1}^\infty \frac{k}{(k+1)!}=1$$

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I can also list many integrals, such as $$\begin{align} \int_0^\frac{\pi}{2}\cos x~dx=1 \end{align}$$ And just for fun: $$0!=1$$ I will try to add some more interesting identities if I can remember/come across them.