The greatest lower bound of $|\cos(x)+\sin(x)|$

Question

I was going through a problem when I stumbled across the need to find the infimum of : $$|\cos(x)+\sin(x)|$$ where $x\in \mathbb R$.

It is easy to get a lower bound, for we must have $|\cos(x)+\sin(x)|>0$ as $|cos(x)|\ge 0$ and $|sin(x)|\ge 0$. But intuitively it makes sense to say that the infimum would be some positive real number, as $\cos(x)$ and $\sin(x)$ won't reduce down to zero for the same $x$.

The existence of the infimum is guaranteed by the completeness of the reals. But how do I find it. Please help. Thanks in advance.

Answer 1

$|\cos x +\sin x|^{2}=1+2\cos x \sin x=1+\sin 2x$. The infimum of $|\cos x +\sin x|^{2}$ is, therfore, $0$. This implies that the infimum of $|\cos x +\sin x|$ is also $0$. It is, in fact, attained at $x=\frac {3\pi} 4$.