Show that there is no complex number such that
`|z|le1/2 and z^(n)sin theta_(0)+z^(n-1)sintheta_(2)+....+zsintheta_(n-1)+ sintheta_(n)=2`
where `theta,theta_(1),theta_(2),……,theta_(n-1), theta_(n)` are reals and ` n in Z^(+)` .

Explanation

we have ,
`|2| = z^(n")sintheta_(0) + z^(n-1)sintheta_(1)+z^(n-2)sintheta_(3) +.......+zsintheta_(n-1)+sintheta_(n)|`
`le 1 + |z| +|z|^(2)+......+ |z|^(n-2)+|z|^(n-1) + |z|^(n), " as" |sintheta_(k)| le 1`
` = (1 - |z|^(n+1))/(1-|z|) " " ( " we assume " |z| lt 1)`
` Rightarrow 1/(1-|z|) - (|z|^(n+1))/(1-|z|) ge2`
`1/(1-|z|) ge 2`
`Rightarrow 1-|z|lt1/2`
`Rightarrow |z|gt1-1/2= 1/2 `
i.e, ` |z| gt 1/2 ` contrary to the given condition ` |z|ge 1/2`
Consequently there does not exist any z with the given condition.