A question is given in my book which I'm unable to solve.
If $\displaystyle\sum_{i=1}^{2022} \sin^{-1}(x_i) = 1011\pi $, then what is the value of $\displaystyle\sum_{i=1}^{2022} x_i$?
Answer of the above problem is given to be $2022$.
In general, this problem is of the form that if $\displaystyle\sum_{i=1}^{n} A_i = k$, then what is the value of $\displaystyle\sum_{i=1}^{n} \sin(A_i).$ So we need to find the sum of sines of numbers, when sum of numbers is given. But I can't continue from here because of my little knowledge.
Alternatively, I thought of using the formula:
$\sin^{-1}(x) + \sin^{-1}(y) = \sin^{-1}(x\sqrt{1-y^2} + y \sqrt{1-x^2})$ to simplify the sum $\displaystyle \sum_{i=1}^{2022} \sin(x_i)$. But here, because of $2022$ terms, it's very difficult to apply the formula.
I think there would be surely any method/formula for simplifying $\sum_{i=1}^n \sin^{-1}(x_i)$ in general, which I'm unaware of.
The range of $\sin^{-1}(x)$, where we assume this is the principal value, is $[-\pi/2,\pi/2]$. The only way for the first sum to equal $1011\pi$ is if all summands have the maximum value of $\pi/2$, which is achieved with $x_i = 1$. Therefore, $\sum_{i=1}^{2022}x_i = 2022$.