Find the minimum and maximum values of $1/(\cos x + \sin x )$

Question

The question states find the maximum and minimum values of $1/(\cos x + \sin x)$ and I turned it into $1/(\sqrt{2} \sin(x+\pi/4))$ and my answers are $1/\sqrt{2}$ for the maximum and $-1/\sqrt{2}$ minimum but the answer is $1/\sqrt{2}$ for the minimum and $-1/\sqrt{2}$ for the maximum.

Why are those the answers?

Thanks

Answer 1

Hint: Let $$f(x)=\frac{1}{\sin(x)+\cos(x)}$$ then $$f'(x)=-{\frac {\cos \left( x \right) -\sin \left( x \right) }{ \left( \sin \left( x \right) +\cos \left( x \right) \right) ^{2}}} $$

Answer 2

Hint: $\sin x + \cos x = \sqrt{2}\sin(x+\pi/4)$. What is the maximal value of this expression? What is the maximal value of this expression? What happens if we invert these extremal values? What happens with the inverse if this value is $0$?