3. Giải các phương trình:

a) ${\sin }^2\frac{x}{2} - 2\cos \frac{x}{2}+2=0;$                    b) 8${\cos }^{2}x + 2\sin x - 7 =0;$

c) 2${\tan }^{2}x + 3\tan x +1=0$;                      d) tanx - 2cotx +1= 0

Giải:

a) ${\sin }^2\frac{x}{2}-2\cos \frac{x}{2}+2=0 \iff 1-{\cos }^2\frac{x}{2}-2\cos \frac{x}{2}+2=0 \iff {\cos }^2\frac{x}{2}+2\cos \frac{x}{2}-3=0$

$\iff {[}_{\cos \frac{x}{2} = -3 \left(loại\right)}^{\cos \frac{x}{2} = 1}\iff \frac{x}{2}=k2\mathrm{\pi }\iff \mathrm{x}=\mathrm{k}4\mathrm{\pi } (\mathrm{k}\in \mathrm{\mathbb{Z}})$

 $$\begin{gathered} b) 8{\cos }^{2}x + 2\sin x - 7 =0 \iff 8(1-{\sin }^{2}x) + 2\sin x - 7=0 \\ \iff 8{\sin }^{2}x - 2\sin x - 1 =0 \iff {[}_{\sin x = -\frac{1}{4}}^{sinx = \frac{1}{2}} \end{gathered}$$

$\iff {[}_{x = arc\sin \left(-\frac{1}{4}\right) + k2\mathrm{\pi }; \mathrm{x} = \mathrm{\pi }-\arcsin \left(-\frac{1}{4}\right)+\mathrm{k}2\mathrm{\pi }}^{x = \frac{\mathrm{\pi }}{6} + k2\mathrm{\pi }; \mathrm{x} = \frac{5\mathrm{\pi }}{6}+\mathrm{k}2\mathrm{\pi }} (k\in \mathrm{\mathbb{Z}})$

c) $2{\tan }^{2}x + 3\tan x + 1 = 0\iff {[}_{\tan x = -\frac{1}{2}}^{\tan x = -1}\iff {[}_{x = arc\tan \left(-\frac{1}{2}\right)-k\mathrm{\pi }}^{x = -\frac{\mathrm{\pi }}{4}+k\mathrm{\pi }}$

d) Đặt t = tanx ta có phương trình t - $\frac{2}{t}+1=0\iff t^2+t-2=0^{}$

$\iff {[}_{t = -2}^{t = 1} \iff {[}_{\tan x = -2}^{\tan x = 1} \iff {[}_{x = arc\tan (-2)+k\mathrm{\pi }}^{x=\frac{\mathrm{\pi }}{4}+k\mathrm{\pi }} (k\in \mathrm{\mathbb{Z}})$