# To me the most common identity that comes to mind that resul

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:
$$\displaystyle{{\cos}^{{2}}\theta}+{{\sin}^{\theta}=}{1}$$ (1)
and maybe
$$\displaystyle-{e}^{{{i}\pi}}={1}$$ (2)
Are there other famous (as in commonly used) identities that yield 1 in particular?

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jgardner33v4
Pick your favorite probability density function f, then
$$\displaystyle{\int_{{-\infty}}^{{\infty}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={1}$$
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Tiefdruckot
$$\displaystyle{\left(\forall{x}\in{\mathbb{{{R}}}}\right)}{{\cos{{h}}}^{{2}}{\left({x}\right)}}-{{\sin{{h}}}^{{2}}{\left({x}\right)}}={1}$$
Vasquez

Here are two well-known infinite series whose sum is 1:
$$\sum_{n=1}^{\infty} \frac{1}{2^n}=\frac12+\frac14+\frac18+\frac{1}{16}+\frac{1}{32}+\ldots=1$$
$$\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\frac{1}{5 \cdot 6}+\ldots=1$$
The following series is less known, interesting although:
$$\sum_{n=1}^{\infty} \frac{1}{s_n}=\frac12+\frac13+\frac17+\frac{1}{43}+\frac{1}{1807}+\ldots=1$$
where denominators form Sylvester sequence: every term is equal to the product of all previous terms, plus one. For example
2
3=2+1
7=2 * 3+1
43=2 * 3 * 7+1
1807=2 * 3 * 7 * 43+1
and so on