What is the least value of co$s^2$θ − 6 cos θ + 12?

Write it as $(cos\theta -3{)}^2+3$. Since cos θ ≤ 1, cos θ − 3 is at least 2 in size, so the least value is $2^2+3=7$ (at cos θ = 1). The vertex value 3 is never reached.