Show that all the roots of equation lies outside the circle

Assume the contrary; i.e. at least one root say z₀ lies on or within the circle |z|=1/2. Then, |z₀| ≤ 1/2

Now,
2 =|z₀ⁿ cosθ₀ + z₀^n-1 cosθ₁ + ... + cosθ_n|
≤ |z₀ⁿ cosθ₀| + |z₀^n-1 cosθ₁| + ... + |cosθ_n| (by triangle's inequality)
≤ |z₀ⁿ| + |z₀^n-1| + ... + 1 (since |cosθ_k ≤1)
≤(1/2)ⁿ + (1/2)^n-1 + (1/2)^n-2 + ... + 1

= 2(1 - (1/2)ⁿ⁺¹)
< 2
which is a contradiction.

As such all roots lie beyond the circle |z|=1/2