Question

If $f\left(x\right)=\sum _{i=1}^{2020}{\cos }^{-1}{\alpha }_i=0$, then $\sum _{i=1}^4{\alpha }_i=$

Answer 1

Given : $\sum _{i=1}^{2020}{\cos }^{-1}{\alpha }_i=0$
We know, $0\le {\cos }^{-1}x\le \pi$
So, given equation will be true only if ${\cos }^{-1}{\alpha }_i=0\text{for}i=1,2,\cdots ,2020$
$\Rightarrow {\alpha }_i=1\text{for}i=1,2,\cdots ,2020$
$\therefore \sum _{i=1}^4{\alpha }_i=1+1+1+1=4$