Cos2x+5=2căn2(2-cosx).sin(x-pi/4)

$$\eqalign{
& \cos 2x + 5 = 2\sqrt 2 \left( {2 – \cos x} \right)\sin \left( {x – {\pi \over 4}} \right) \cr
& \Leftrightarrow {\cos ^2}x – {\sin ^2}x + 5 = 2\left( {2 – \cos x} \right)\left( {\sin x – \cos x} \right) \cr
& \Leftrightarrow \left( {\cos x – \sin x} \right)\left( {\cos x + \sin x} \right) + 5 = 2\left( {2 – \cos x} \right)\left( {\sin x – \cos x} \right) \cr
& \Leftrightarrow \left( {\cos x – \sin x} \right)\left( {\cos x + \sin x + 4 – 2\cos x} \right) + 5 = 0 \cr
& \Leftrightarrow \left( {\cos x – \sin x} \right)\left( {\sin x – \cos x + 4} \right) + 5 = 0 \cr
& \Leftrightarrow – {\left( {\cos x – \sin x} \right)^2} + 4\left( {\cos x – \sin x} \right) + 5 = 0 \cr
& \Leftrightarrow \left[ \matrix{
\cos x – \sin x = – 1 \hfill \cr
\cos x – \sin x = 5\,\,\left( {Vo\,\,nghiem} \right) \hfill \cr} \right. \cr
& \Leftrightarrow \sqrt 2 \cos \left( {x + {\pi \over 4}} \right) = – 1 \cr
& \Leftrightarrow \cos \left( {x + {\pi \over 4}} \right) = – {1 \over {\sqrt 2 }} \cr
& \Leftrightarrow \left[ \matrix{
x + {\pi \over 4} = {{3\pi } \over 4} + k2\pi \hfill \cr
x + {\pi \over 4} = – {{3\pi } \over 4} + k2\pi \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
x = {\pi \over 2} + k2\pi \hfill \cr
x = – \pi + k2\pi \hfill \cr} \right.\,\,\left( {k \in Z} \right) \cr} $$