If ${\displaystyle \sum _{i=1}^{10}\cos {\theta }_i}=10$   then $\sin {\theta }_1+\sin {\theta }_2+\sin {\theta }_3+----+\sin {\theta }_{10}=$

$Cos{\theta }_1+Cos{\theta }_2+Cos{\theta }_3+---+Cos{\theta }_{10}=10$

Since maximum value of $Cos\theta =1$     

  $\Rightarrow$   ${\theta }_1={\theta }_2={\theta }_3=---={\theta }_{10}=0.$

$\therefore Sin{\theta }_1+Sin{\theta }_2+Sin{\theta }_3+---+Sin{\theta }_{10}=0.$