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?

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.
