<s>[INST] Solve $\left ( 1- \sqrt{2}\sin x \right )\left ( \cos 2x+ \sin 2x \right )= \frac{1}{2}$ Solve 
$$\left ( 1- \sqrt{2}\sin x \right )\left ( \cos 2x+ \sin 2x \right )=  \frac{1}{2}$$
Now I did not understand how can i solve that.
I have tried substituting  $\cos(2x)=\cos^2(x)−\sin^2(x)$ and$\,$ $\sin(2x)=2\sin(x)\cos(x)$,
the equation is now $(1−\sqrt2\sin(x))(\cos^2(x)−\sin^2(x)+2\sin(x)\cos(x))=\frac12$
Help Required
Thanks
 [/INST] Multiplying by $1+\sqrt 2 \sin x$ on both sides and using $1-2 \sin^2 x = \cos 2x$ we get
$\cos 2x (\cos 2x + \sin 2x ) = \dfrac{1+\sqrt 2 \sin x}{2}$
or $1+\cos 4x + \sin 4x = 1+\sqrt 2 \sin x$
or $\dfrac{\sin 4 x + \cos 4x}{\sqrt 2} = \sin x$
or $\sin \left(4x+\dfrac{\pi}{4} \right) = \sin x$ which can be easily solved
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