A geometry problem by Ossama Ismail

Relevant wiki: Trigonometric Equations - R method

$\begin{aligned} \sin(x) + \cos(x) &= \sqrt 2 \\ \dfrac{1}{\sqrt 2} \sin(x) + \dfrac{1}{\sqrt 2} \cos(x) &= 1 \\ \\ \text{The above equation can be written as } \\ \\ \cos(\theta) \sin(x) + \sin(\theta) \cos(x) &= 1 \\ \text{where } \ \ \theta = \sin^{-1}(\dfrac{1}{\sqrt 2}) &= 45^\circ \\ \\ \text {then} \ \ \sin(x + \theta) &= 1 \\ x+ 45 &= \sin^{-1}(1)\\ x+ 45 &= 90^\circ \\ x &= 45^\circ \\ \end{aligned}$