Question

Solve $\cos 2\theta =(\sqrt{2}+1)(\cos \theta -1\sqrt{2})$.

Answer 1

If $\cos \theta =c$, then
$2c^2-1=\frac{\sqrt{2}+1}{\sqrt{2}}(\sqrt{2}c-1)$
or $(\sqrt{2}c-1)\left[\right(\sqrt{2}c+1)-(1+\frac{1}{\sqrt{2}})]=0$
or $(\sqrt{2}c-1)\sqrt{2}(c-\frac{1}{2})=0$
$\therefore c=\frac{1}{\sqrt{2}},\frac{1}{2}$
$\therefore \theta =2n\pi \pm \frac{\pi }{4}$ or $2n\pi \pm \frac{\pi }{3}$.