<s>[INST] 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?
 [/INST] 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.
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