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

PEEWSRIGWETRYqx 2022-01-01 Answered
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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Expert Answer

jgardner33v4
Answered 2022-01-02 Author has 3902 answers
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
Answered 2022-01-03 Author has 2049 answers
\(\displaystyle{\left(\forall{x}\in{\mathbb{{{R}}}}\right)}{{\cos{{h}}}^{{2}}{\left({x}\right)}}-{{\sin{{h}}}^{{2}}{\left({x}\right)}}={1}\)
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Vasquez
Answered 2022-01-08 Author has 8850 answers

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

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