$\sqrt{3}\sin x$ + 2$\cos x$ - $\cos 2x$ -1 = 0
$\sqrt3sinx+2cosx-cos2x-1=0$
$\Leftrightarrow \sqrt3sinx+2cosx-2cos^2x=0$
$\Leftrightarrow \sqrt3sinx=2cos^2x-2cosx=2cosx(cosx-1)    (*)$
$\Rightarrow 3sin^2x=4cos^2x(cosx-1)^2$
$\Rightarrow 3(1-cos^2x)=4cos^2x(cosx-1)^2$
$\Rightarrow (cosx-1)[4cos^2x(cosx-1)+3(cosx+1)]=0$
$\Rightarrow (cosx-1)(4cos^3x-4cos^2x+3cosx+3)=0$
$\Rightarrow (cosx-1)(2cosx+1)(2cos^2x-3cosx+3)=0$
$\Rightarrow cosx=1,\frac{-1}{2}$
Thay vào  $(*)$ ta tính được
$\begin{cases}cosx=1 \\ sinx=0 \end{cases}$  hoặc $\begin{cases}cosx=-\frac{1}{2} \\ sinx=\frac{\sqrt3}{2} \end{cases}$
$\Leftrightarrow x=2k\pi , \frac{2\pi}{3}+2k\pi$
Dễ hiểu lắm! :D –  lehoa1503 26-06-13 04:59 PM
Hay quá! ^^ –  phan.le.thanh.hien.10.01.hht 26-06-13 12:08 PM
Cách này hay quá! Mình ít khi nghĩ tới phương trình hệ quả. :) –  binhnguyenhoangvu 26-06-13 11:50 AM
$\sqrt 3 \sin x + 2\cos x - \cos 2x - 1 = 0$
$ \Leftrightarrow \sqrt 3 \sin x + 2\cos x - {\cos ^2}x + {\sin ^2}x - 1 = 0$
$ \Leftrightarrow \sqrt 3 \sin x + {\sin ^2}x - ({\cos ^2}x - 2\cos x + 1) = 0$
$\Leftrightarrow \sin x(\sin x + \sqrt 3 ) - {(1 - \cos x)^2} = 0$
$ \Leftrightarrow 2\sin \frac{x}{2}\cos \frac{x}{2}(\sin x + \sqrt 3 ) - 2{\sin ^2}\frac{x}{2}(1 - \cos x) = 0$
$ \Leftrightarrow \sin \frac{x}{2} = 0 \vee \cos \frac{x}{2}(\sin x + \sqrt 3 ) - \sin \frac{x}{2}(1 - \cos x) = 0$
$*\sin \frac{x}{2} = 0 \Leftrightarrow x = k2\pi   (k \in Z)$
$*\cos \frac{x}{2}(\sin x + \sqrt 3 ) - \sin \frac{x}{2}(1 - \cos x) = 0$
$ \Leftrightarrow \sin x\cos \frac{x}{2} + \cos x\sin \frac{x}{2} - \sin \frac{x}{2} + \sqrt 3 \cos \frac{x}{2} = 0$
$\Leftrightarrow \sin \frac{{3x}}{2} - 2\sin (\frac{x}{2} - \frac{\pi }{3}) = 0  (2)$
Đặt $t = \frac{x}{2} - \frac{\pi }{3} \Rightarrow \frac{{3x}}{2} = 3t + \pi  \Rightarrow \sin \frac{{3x}}{2} = \sin (3t + \pi ) = - \sin 3t$
$(2) \Leftrightarrow  - \sin 3t - 2\sin t = 0 \Leftrightarrow 4{\sin ^3}t - 5\sin t = 0$
$ \Leftrightarrow {\sin ^2}t = \frac{5}{4}(VL) \vee \sin t = 0 \Leftrightarrow t = \frac{x}{2} - \frac{\pi }{3} = h\pi  \Leftrightarrow x = \frac{{2\pi }}{3} + h2\pi (h \in Z)$

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