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: cos2θ+sinθ=1 (1) and maybe −eiπ=1 (2) Are there other famous (as in commonly used) identities that yield 1 in particular?

Name: 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: cos2θ+sinθ=1 (1) and maybe −eiπ=1 (2) Are there other famous (as in commonly used) identities that yield 1 in particular?
Uploaded: 2022-02-23T17:22:57+00:00
Description: 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: cos2θ+sinθ=1 (1) and maybe −eiπ=1 (2) Are there other famous (as in commonly used) identities that yield 1 in particular?

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:  cos2θ+sinθ=1     (1)  and maybe  −eiπ=1        (2)  Are there other famous (as in commonly used) identities that yield 1 in particular?

jgardner33v4 · Accepted Answer

Pick your favorite probability density function f, then  ∫−∞∞f(x)dx=1

Tiefdruckot · Accepted Answer

(∀x∈R)cos⁡h2(x)−sin⁡h2(x)=1

Vasquez · Accepted Answer

Here are two well-known infinite series whose sum is 1: ∑n=1∞12n=12+14+18+116+132+…=1 ∑n=1∞1n(n+1)=11⋅2+12⋅3+13⋅4+14⋅5+15⋅6+…=1 The following series is less known, interesting although: ∑n=1∞1sn=12+13+17+143+11807+…=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