Question

The number of solution(s) of the equation $\cos 2\theta =(\sqrt{2}+1)(\cos \theta -\frac{1}{\sqrt{2}})$, in the interval $(-\frac{\pi }{4},\frac{3\pi }{4})$, is

• A. $4$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

$\sqrt{2}\cos 2\theta =(\sqrt{2}+1)(\sqrt{2}\cos \theta -1)\Rightarrow \sqrt{2}\cos 2\theta =2\cos \theta +\sqrt{2}\cos \theta -\sqrt{2}-1\Rightarrow 2\cos \theta +\sqrt{2}\cos \theta -\sqrt{2}-1=\sqrt{2}(2{\cos }^2\theta -1)\Rightarrow 2\sqrt{2}{\cos }^2\theta -(2+\sqrt{2})\cos \theta +1=0\Rightarrow (2\cos \theta -1)(\sqrt{2}\cos \theta -1)=0$
$\Rightarrow \cos \theta =\frac{1}{2}$ or $\cos \theta =\frac{1}{\sqrt{2}}$

$\theta =\frac{\pi }{3},\frac{5\pi }{3},\frac{\pi }{4},\frac{7\pi }{4}$