Giải phương trình:  1/cosx -1 1/sinx = 2 căn 2 cos (x+pi/4)

Trả lời

$\frac{1}{\cos x}-\frac{1}{\sin x}=2\sqrt{2}\cos \left(x+\frac{\pi }{4}\right)$

$\iff \frac{\sin x-\cos x}{\sin x.\cos x}=2\sqrt{2}\left(\cos x.\cos \frac{\pi }{4}-\sin x.\sin \frac{\pi }{4}\right)$

$\iff \frac{\sin x-\cos x}{2\sin x.\cos x}=\sqrt{2}\left(\cos x.\frac{\sqrt{2}}{2}-\sin x.\frac{\sqrt{2}}{2}\right)$

$\iff \frac{\sin x-\cos x}{\sin 2x}=\cos x-\sin x$

$\iff \left(\sin x-\cos x\right)\left(\frac{1}{\sin 2x}+1\right)=0$

$\iff \left[\begin{array}{l}\sin x=\cos x\\ \frac{1}{\sin 2x}=-1\end{array}\right.\iff \left[\begin{array}{l}\tan x=1\\ \sin 2x=-1\end{array}\right.$

$\iff \left[\begin{array}{l}x=\frac{\pi }{4}+k\pi \\ 2x=-\frac{\pi }{2}+k2\pi \end{array}\right.\left(k\in \mathbb{Z}\right)\iff \left[\begin{array}{l}x=\frac{\pi }{4}+k\pi \\ x=-\frac{\pi }{4}+k\pi \end{array}\right.\left(k\in \mathbb{Z}\right)$

$\iff x=\frac{\pi }{4}+\frac{k\pi }{2}\left(k\in \mathbb{Z}\right)$