$2{{\operatorname{Sin}}^{2}}(x-\frac{\pi }{8})+3\operatorname{Cos}(x-\frac{5\pi }{8})=5\Rightarrow 2{{\operatorname{Sin}}^{2}}(x-\frac{\pi }{8})+3\operatorname{Cos}((x-\frac{\pi }{8})-\frac{\pi }{2})-5=0$ 

می‌دانیم $\operatorname{Cos}(\alpha -\frac{\pi }{2})=\operatorname{Cos}(\frac{\pi }{2}-\alpha )=\operatorname{Sin}\alpha $. بنابراین:

$2{{\operatorname{Sin}}^{2}}(x-\frac{\pi }{8})+3\operatorname{Sin}(x-\frac{\pi }{8})-5=0\Rightarrow \operatorname{Sin}(x-\frac{\pi }{8})=1*\operatorname{Sin}(x-\frac{\pi }{8})=\frac{c}{a}=-\frac{5}{2}$ 

 واضح است که $\operatorname{Sin}(x-\frac{\pi }{8})=-\frac{5}{2}$ غیرممکن است. پس:

$\operatorname{Sin}(x-\frac{\pi }{8})=1\to x-\frac{\pi }{8}=2k\pi +\frac{\pi }{2}\Rightarrow x=2k\pi +\frac{\pi }{2}+\frac{\pi }{8}\Rightarrow x=2k\pi +\frac{5\pi }{8}\Rightarrow \frac{k}{x}=\frac{0}{\frac{5\pi }{8}}$ 

$x=\frac{5\pi }{8}$ تنها جواب معادله در بازه‌ی $\left[ 0,2\pi  \right]$ است.