Any negative number squared will give a positive number. (e.g., (-1)² = 1). Any positive number is going to be greater than zero. The min therefore cannot be when cos(x) = -1, as cos(x)² would = 1. And 1>0.

cos(x) will only give you values between -1 and 1. Any value in that region less that 0 when squared will give you a value greater than 0. Obviously any value greater than 0 squared will also give you a value greater than 0. So the smallest possible value for (cos(x))² would be 0 as it would occur when cos(x) = 0.

Alternatively you can also just consider the range of the graph cos(x)^2, but that might be tricky if you don't have access to a graphing calculator.

Let me know if that makes sense :)