Answer
The statement is flawed. Taking the square root of each side cannot give $\sin\theta+\cos\theta=1$ as wished.
Work Step by Step
(Since $\sin^2\theta+\cos^2\theta=1$, ...)
This part is correct.
(... I should be able to also say that $\sin\theta+\cos\theta=1$ if I take the square root of each side.)
However, if we take the square root of each side, that means
$$\sqrt{\sin^2\theta+\cos^2\theta}=\sqrt1=1$$
Unfortunately,
$$\sqrt{\sin^2\theta+\cos^2\theta}\ne\sqrt{\sin^2\theta}+\sqrt{\cos^2\theta}=|\sin\theta|+|\cos\theta|$$
And also, $$|\sin\theta|+|\cos\theta|\ne\sin\theta+\cos\theta$$
Therefore,
$$\sqrt{\sin^2\theta+\cos^2\theta}\ne\sin\theta+\cos\theta\ne1$$
Consequently, what the student has said is wrong. Taking the square root of each side cannot give $\sin\theta+\cos\theta=1$ as wished.
