Calculation of $left(frac{1}{cos^2x}right)^{frac{1}{2}}$

You are correct that $sqrt{x^2} neq x$ for every $xinBbb R$. Otherwise, we would have absurdity like $1 = sqrt{1}=sqrt{(-1)^2} =-1$. For $xinBbb R$ we indeed have $sqrt{x^2}=|x|$. However, we have $sqrt{x^2}=x$ for all $xgeq 0$.

Hence, as you professor seems to assume that $tin [-pi/2,pi/2]$, his claim that $sqrt{frac{1}{cos^2(x)}}=sec(t)$ is correct. Indeed for $tin [-pi/2,pi/2]$, we have $cos(t)geq 0$.