Famous or common mathematical identities that yield $1$ 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?
 A: Pick your favorite probability density function $f$, then
$$
\int_{-\infty}^\infty f(x)dx=1.
$$
A: $$(\forall x\in \Bbb R)\;\;\cosh^2(x)-\sinh^2(x)=1$$
A: $$\sec^2x-\tan^2x = 1$$
$$\csc^2x-\cot^2x = 1$$
$$\sin x \csc x = 1$$
$$\tan x\cot x = 1$$
$$\sec x\cos x = 1$$
$$ \frac{\sin x + \cos x}{\cos ^3x} - \tan^3x - \tan^2x-\tan x = 1$$
A: 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:
$$\sum_{k=1}^\infty \frac{k}{(k+1)!}=1$$

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.
A: 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.
