a/

\(\Leftrightarrow3cos^2x-4sinx.cosx+1-cos^2x=1\)

\(\Leftrightarrow2cos^2x-4sinx.cosx=0\)

\(\Leftrightarrow2cosx\left(cosx-2sinx\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\tanx=\frac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{2}+k\pi\\x=arctan\left(\frac{1}{2}\right)+k\pi\end{matrix}\right.\)

b.

Nhận thấy \(cosx=0\) ko phải nghiệm, chia 2 vế cho \(cos^2x\)

\(4-3tanx+3tan^2x=1+tan^2x\)

\(\Leftrightarrow2tan^2x-3tanx+3=0\)

Pt vô nghiệm

TH1: Xét cox = 0 ( có p là nghiệm ko)

TH2: Xét \(\cos x\ne0\). Ta chia cả hai vế \(\cos^2x\)

Pt trở thành \(2\tan^2x-4\tan x+4-1\left(1+\tan^2x\right)=0\)

\(\Leftrightarrow\tan^2x-4\tan x+3=0\)

Đặt \(\tan x=t\). Giải pt nữa là xg ạ

\(2sin^2x-4sinx.cosx+4cos^2x=1\)

\(\Leftrightarrow2\left(sin^2x+cos^2x\right)-4sinx.cosx+2cos^2x-1=0\)

\(\Leftrightarrow2-2sin2x+cos2x=0\)

\(\Leftrightarrow2sin2x-cos2x=2\)

\(\Leftrightarrow\sqrt{5}\left(\dfrac{2}{\sqrt{5}}sin2x-\dfrac{1}{\sqrt{5}}cos2x\right)=2\)

\(\Leftrightarrow sin\left(2x-arccos\dfrac{2}{\sqrt{5}}\right)=\dfrac{2}{\sqrt{5}}\)

\(\Leftrightarrow\left[{}\begin{matrix}2x-arccos\dfrac{2}{\sqrt{5}}=arcsin\dfrac{2}{\sqrt{5}}+k2\pi\\2x-arccos\dfrac{2}{\sqrt{5}}=\pi-arcsin\dfrac{2}{\sqrt{5}}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{1}{2}arccos\dfrac{2}{\sqrt{5}}+\dfrac{1}{2}arcsin\dfrac{2}{\sqrt{5}}+k\pi\\x=\dfrac{\pi}{2}+\dfrac{1}{2}arccos\dfrac{2}{\sqrt{5}}-\dfrac{1}{2}arcsin\dfrac{2}{\sqrt{5}}+k\pi\end{matrix}\right.\)

Pt \(\Leftrightarrow\left(2sinx-1\right)\left(2sin2x-1\right)=3-4\left(1-sin^2x\right)\)

\(\Leftrightarrow2sin2x\left(2sinx-1\right)-2sinx+1=-1+4sin^2x\)

\(\Leftrightarrow2sin2x\left(2sinx-1\right)-\left(4sin^2x+2sinx-2\right)=0\)

\(\Leftrightarrow2sin2x\left(2sinx-1\right)-2\left(2sinx-1\right)\left(sinx+1\right)=0\)

\(\Leftrightarrow2\left(2sinx-1\right)\left(sin2x-sinx-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=\dfrac{1}{2}\left(1\right)\\sin2x=sinx+1\left(2\right)\end{matrix}\right.\)

Từ (1) \(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+k2\pi\\x=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\),\(k\in Z\)

Từ (2)\(\Leftrightarrow2sinx.cosx-sinx-1=0\)

(Cái này tạm thời nghĩ ko ra,tối làm :)

\(y=sin\left(\frac{\pi}{2}-2x\right)=cos2x\Rightarrow y'=-2sin2x\)

1.

\(3sin^22x-2sin2x.cos2x-4cos^22x=2\)

\(\Leftrightarrow-\dfrac{3}{2}\left(1-2sin^22x\right)-2sin2x.cos2x-2\left(2cos^22x-1\right)=\dfrac{5}{2}\)

\(\Leftrightarrow sin4x+\dfrac{7}{2}cos4x=-\dfrac{5}{2}\)

\(\Leftrightarrow\dfrac{\sqrt{53}}{2}\left(\dfrac{2}{\sqrt{53}}sin4x+\dfrac{7}{\sqrt{53}}cos4x\right)=-\dfrac{5}{2}\)

\(\Leftrightarrow sin\left(4x+arccos\dfrac{2}{\sqrt{53}}\right)=-\dfrac{5}{\sqrt{53}}\)

\(\Leftrightarrow\left[{}\begin{matrix}4x+arccos\dfrac{2}{\sqrt{53}}=arcsin\left(-\dfrac{5}{\sqrt{53}}\right)+k2\pi\\4x+arccos\dfrac{2}{\sqrt{53}}=\pi-arcsin\left(-\dfrac{5}{\sqrt{53}}\right)+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{1}{4}arccos\dfrac{2}{\sqrt{53}}+\dfrac{1}{4}arcsin\left(-\dfrac{5}{\sqrt{53}}\right)+\dfrac{k\pi}{2}\\x=\dfrac{\pi}{4}-\dfrac{1}{4}arccos\dfrac{2}{\sqrt{53}}-\dfrac{1}{4}arcsin\left(-\dfrac{5}{\sqrt{53}}\right)+\dfrac{k\pi}{2}\end{matrix}\right.\)

2.

\(2\sqrt{3}cos^2x+6sinx.cosx=3+\sqrt{3}\)

\(\Leftrightarrow\sqrt{3}\left(2cos^2x-1\right)+6sinx.cosx=3\)

\(\Leftrightarrow\sqrt{3}cos2x+3sin2x=3\)

\(\Leftrightarrow2\sqrt{3}\left(\dfrac{1}{2}cos2x+\dfrac{\sqrt{3}}{2}sin2x\right)=3\)

\(\Leftrightarrow cos\left(2x-\dfrac{\pi}{3}\right)=\dfrac{\sqrt{3}}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}2x-\dfrac{\pi}{3}=\dfrac{\pi}{6}+k2\pi\\2x-\dfrac{\pi}{3}=-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k\pi\\x=\dfrac{\pi}{12}+k\pi\end{matrix}\right.\)

3.

\(3cos^24x+5sin^24x=2-2\sqrt{3}sin4x.cos4x\)

\(\Leftrightarrow4cos^24x+4sin^24x-cos^24x+sin^24x=2-2\sqrt{3}sin4x.cos4x\)

\(\Leftrightarrow4-cos8x=2-\sqrt{3}sin8x\)

\(\Leftrightarrow cos8x-\sqrt{3}sin8x=2\)

\(\Leftrightarrow\dfrac{1}{2}cos8x-\dfrac{\sqrt{3}}{2}sin8x=1\)

\(\Leftrightarrow cos\left(8x+\dfrac{\pi}{3}\right)=1\)

\(\Leftrightarrow8x+\dfrac{\pi}{3}=k2\pi\)

\(\Leftrightarrow x=-\dfrac{\pi}{24}+\dfrac{k\pi}{4}\)

a, (sinx + cosx)(1 - sinx . cosx) = (cosx - sinx)(cosx + sinx)

⇔ \(\left[{}\begin{matrix}sinx+cosx=0\\cosx-sinx=1-sinx.cosx\end{matrix}\right.\)

⇔ \(\left[{}\begin{matrix}sinx+cosx=0\\cosx+sinx.cosx-1-sinx=0\end{matrix}\right.\)

⇔ \(\left[{}\begin{matrix}sinx+cosx=0\\\left(cosx-1\right)\left(sinx+1\right)=0\end{matrix}\right.\)

⇔ \(\left[{}\begin{matrix}sin\left(x+\dfrac{\pi}{4}\right)=0\\cosx=1\\sinx=-1\end{matrix}\right.\)

b, (sinx + cosx)(1 - sinx . cosx) = 2sin2x + sinx + cosx

⇔ (sinx + cosx)(1 - sinx.cosx - 1) = 2sin2x

⇔ (sinx + cosx).(- sinx . cosx) = 2sin2x

⇔ 4sin2x + (sinx + cosx) . sin2x = 0

⇔ \(\left[{}\begin{matrix}sin2x=0\\\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)+4=0\end{matrix}\right.\)

⇔ sin2x = 0

c, 2cos³x = sin3x

⇔ 2cos³x = 3sinx - 4sin³x

⇔ 4sin³x + 2cos³x - 3sinx(sin²x + cos²x) = 0

⇔ sin³x + 2cos³x - 3sinx.cos²x = 0

Xét cosx = 0 : thay vào phương trình ta được sinx = 0. Không có cung x nào có cả cos và sin = 0 nên cosx = 0 không thỏa mãn phương trình

Xét cosx ≠ 0 chia cả 2 vế cho cos³x ta được : 

tan³x + 2 - 3tanx = 0

⇔ \(\left[{}\begin{matrix}tanx=1\\tanx=-2\end{matrix}\right.\)

d, cos²x - \(\sqrt{3}sin2x\) = 1 + sin²x

⇔ cos²x - sin²x - \(\sqrt{3}sin2x\) = 1

⇔ cos2x - \(\sqrt{3}sin2x\) = 1

⇔ \(2cos\left(2x+\dfrac{\pi}{3}\right)=1\)

⇔ \(cos\left(2x+\dfrac{\pi}{3}\right)=\dfrac{1}{2}=cos\dfrac{\pi}{3}\)

e, cos³x + sin³x = 2cos⁵x + 2sin⁵x

⇔ cos³x (1 - 2cos²x) + sin³x (1 - 2sin²x) = 0

⇔ cos³x . (- cos2x) + sin³x . cos2x = 0

⇔ \(\left[{}\begin{matrix}sin^3x=cos^3x\\cos2x=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}sinx=cosx\\cos2x=0\end{matrix}\right.\)

⇔ \(\left[{}\begin{matrix}sin\left(x-\dfrac{\pi}{4}\right)=0\\cos2x=0\end{matrix}\right.\)