sin & cos... Can you find the answer?

First, we must clarify that $x$ is in degree . Let $f(x) = \sin x + \cos x$ and $f(x)$ will have maximum value for $0^\circ \le x \le 90^\circ$ . In order to obtain the maximum value of $f(x)$ we use the standard method: $f'(x)=0$ and $f''(x)<0$ . We obtain: $f'(x) = \cos x - \sin x = 0$ and $f''(x) = -\sin x - \cos x < 0.$ Thus, $\begin{aligned} f'(x) &= 0 \\ \cos x - \sin x &= 0\\ \tan x &= 1\\ x &= \boxed{45^\circ} \end{aligned}$ $$ $\text{\# }\mathbb{Q}.\mathbb{E}.\mathbb{D}.\text{\#}$