Giải các phương trình sau:
a) $\sin \left(2x-\frac{\pi }{6}\right)=-\frac{\sqrt{3}}{2}$ b) $\cos \left(\frac{3x}{2}+\frac{\pi }{4}\right)=\frac{1}{2}$ c) $\sin 3x-\cos 5x=0$ d) ${\cos }^{2}x=\frac{1}{4}$

e) $\sin x-\sqrt{3}\cos x=0$ f) $\sin x+\cos x=0$ g) $\tan 3x=-\frac{1}{\sqrt{3}}$ h) $\tan \left(2x+10^0\right)=\tan 60^0$

k) $\cot 4x=\sqrt{3}$ l) $\cot \left(x+2\right)=1$.

(AI giải đề cho tôi)
Câu 10. Giải từng phương trình lượng giác cơ bản:

a) $\sin \left(2x-\frac{\pi }{6}\right)=-\frac{\sqrt{3}}{2}$
Ta có $2x-\frac{\pi }{6}=-\frac{\pi }{3}+k2\pi$ hoặc $2x-\frac{\pi }{6}=\pi +\frac{\pi }{3}+k2\pi =-\frac{2\pi }{3}+k2\pi$ (do $-\frac{2\pi }{3}$ tương đương $\frac{4\pi }{3}$).
Suy ra $x=-\frac{\pi }{12}+k\pi$ hoặc $x=-\frac{\pi }{4}+k\pi$, $k\in \mathbb{Z}$.

b) $\cos \left(\frac{3x}{2}+\frac{\pi }{4}\right)=\frac{1}{2}$
$\frac{3x}{2}+\frac{\pi }{4}=\pm \frac{\pi }{3}+k2\pi$.
Trường hợp $+$: $\frac{3x}{2}=\frac{\pi }{12}+k2\pi \Rightarrow x=\frac{\pi }{18}+\frac{4k\pi }{3}$.
Trường hợp $-$: $\frac{3x}{2}=-\frac{7\pi }{12}+k2\pi \Rightarrow x=-\frac{7\pi }{18}+\frac{4k\pi }{3}$, $k\in \mathbb{Z}$.

c) $\sin 3x-\cos 5x=0\iff \sin 3x=\cos 5x=\sin \left(\frac{\pi }{2}-5x\right)$
$\iff 2\cos \left(\frac{3x+\frac{\pi }{2}-5x}{2}\right)\sin \left(\frac{3x-\frac{\pi }{2}+5x}{2}\right)=0$
$\iff \cos \left(\frac{\pi }{4}-x\right)\sin \left(4x-\frac{\pi }{4}\right)=0$.
Từ đó $x=\frac{\pi }{4}+k\pi$ hoặc $4x=\frac{\pi }{4}+k\pi \Rightarrow x=\frac{\pi }{16}+\frac{k\pi }{4}$, $k\in \mathbb{Z}$.

d) ${\cos }^{2}x=\frac{1}{4}\iff \frac{1+\cos 2x}{2}=\frac{1}{4}\iff \cos 2x=-\frac{1}{2}$
$\iff 2x=\pm \frac{2\pi }{3}+k2\pi \iff x=\pm \frac{\pi }{3}+k\pi$, $k\in \mathbb{Z}$.

e) $\sin x-\sqrt{3}\cos x=0$. Vì $\cos x\ne 0$ (nếu $\cos x=0$ thì $\sin x=0$, vô lý), chia hai vế cho $\cos x$:
$\tan x=\sqrt{3}\iff x=\frac{\pi }{3}+k\pi$, $k\in \mathbb{Z}$.

f) $\sin x+\cos x=0\iff \tan x=-1$ (tương tự $\cos x\ne 0$)
$\iff x=-\frac{\pi }{4}+k\pi$, $k\in \mathbb{Z}$.

g) $\tan 3x=-\frac{1}{\sqrt{3}}\iff 3x=-\frac{\pi }{6}+k\pi \iff x=-\frac{\pi }{18}+\frac{k\pi }{3}$, $k\in \mathbb{Z}$.

h) $\tan \left(2x+10^{\circ }\right)=\tan 60^{\circ }\iff 2x+10^{\circ }=60^{\circ }+k\cdot 180^{\circ }$
$\iff x=25^{\circ }+k\cdot 90^{\circ }$, $k\in \mathbb{Z}$.

k) $\cot 4x=\sqrt{3}\iff 4x=\frac{\pi }{6}+k\pi \iff x=\frac{\pi }{24}+\frac{k\pi }{4}$, $k\in \mathbb{Z}$.

l) $\cot (x+2)=1\iff x+2=\frac{\pi }{4}+k\pi \iff x=\frac{\pi }{4}-2+k\pi$, $k\in \mathbb{Z}$ (điều kiện $\sin (x+2)\ne 0$ được thỏa vì $\cot =1$).