Giải phương trình:    $\sqrt{\tan x+\sin x}+\sqrt{\tan x-\sin x}=2\sqrt{\tan x}.\cos x$
Điều kiện:  $\begin{cases}\tan x\geq |\sin x| \\ \cos x\neq 0 \end{cases}$
Nhận thấy $\cos x<0$ không thỏa mãn phương trình. Xét $\cos x>0$
Phương trình tương đương:

              $(\sqrt{\tan x+\sin x}+\sqrt{\tan x-\sin x})^2=4\tan x.\cos^2x$
 $\Leftrightarrow       2\tan x+2\sqrt{\tan ^2x-\sin ^2x}=4\tan x.\cos^2x$
 $\Leftrightarrow       \sqrt{\tan ^2x-\sin ^2x}=\tan x(2\cos^2 x-1)$

$\Leftrightarrow      \begin{cases}\tan ^2x-\sin ^2x=\tan^2 x(2\cos^2 x-1)^2 \\ \tan x(2\cos^2 x-1)\geq 0 \end{cases}$

$\Leftrightarrow       \begin{cases}\sin ^2x(1-\cos ^2x)=\sin ^2x(2\cos ^2x-1)^2\\ \tan x\cos 2x\geq 0\end{cases}$

 $\Leftrightarrow    \begin{cases}  \left[ \begin{array}{l} \sin x = 0\\1-\cos ^2x=4\cos ^4x-4\cos ^2x+1 \end{array} \right.  \\ \tan x\geq 0, \cos 2x\geq 0 \end{cases}.$

$\Leftrightarrow     \begin{cases} \left[ \begin{array}{l}x = k\pi\\\cos x = \frac{\sqrt{3}}{2} ( do \cos x>0)\end{array} \right.\\ \tan x\geq 0\end{cases}         \Leftrightarrow       \left[ \begin{array}{l}x = k\pi\\x =\frac{\pi}{6}+k2\pi\end{array} \right.$
 So với điều kiện, ta có nghiệm cần tìm là         $\left[ \begin{array}{l}x = k\pi\\x =\frac{\pi}{6}+k2\pi \end{array} \right.            (k\in Z)$

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