Giải các phương trình :

a) \(\cos^2x+\cos^22x-\cos^23x-\cos^24x=0\)

b) \(\cos4x\cos\left(\pi+2x\right)-\sin2x\cos\left(\dfrac{\pi}{2}-4x\right)=\dfrac{\sqrt{2}}{2}\sin4x\)

c) \(\tan\left(120^0+3x\right)-\tan\left(140^0-x\right)=2\sin\left(80^0+2x\right)\)

d) \(\tan^2\dfrac{x}{2}+\sin^2\dfrac{x}{2}\tan\dfrac{x}{2}+\cos^2\dfrac{x}{2}+\cot^2\dfrac{x}{2}+\sin x=4\)

e) \(\dfrac{\sin2t+2\cos^2t-1}{\cot t-\cot3t+\sin3t-\sin t}=\cos t\)

\(cosx-2cos3x=1+\sqrt{3}sinx\)

\(sinx+sinx\left(x+\dfrac{\pi}{3}\right)+sin4x=sin\left(2x-\dfrac{\pi}{3}\right)\)

\(\left(1-\dfrac{1}{2sinx}\right)cos^22x=2sinx-3+\dfrac{1}{sinx}\)

( sinx -2cosx)cos2x + sinx = (cos4x - 1)cosx +\(\dfrac{cos2x}{2sinx}\)

\(\left(\dfrac{cos4x+sin2x}{cos3x+sin3x}\right)^2=2\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)+3\)

a, cos4x + 12sin²x -1 = 0

b, cos⁴x - sin⁴x + cos4x = 0

c, 5.(sinx + \(\dfrac{cos3x+sin3x}{1+2sin2x}\) ) = 3 + cos2x với mọi x\(\in\left(0;2\pi\right)\)

d, \(\dfrac{sin3x}{3}=\dfrac{sin5x}{5}\)

e, \(\dfrac{sin5x}{5sinx}=1\)

f, cos²3x - cos2x - cos²x =0

g, cos⁴x + sin⁴x + cos(\(x-\dfrac{\pi}{4}\) ) . sin(\(3x-\dfrac{\pi}{4}\) ) - \(\dfrac{3}{2}\) = 0

h, sin\(\left(2x+\dfrac{5\pi}{2}\right)\) - 3cos\(\left(x-\dfrac{7\pi}{2}\right)\)= 1 + 2sinx với x\(\in\left(\dfrac{\pi}{2};2\pi\right)\)

i, 5sinx - 2 = 3.( 1- sinx ) . tan3x

k, ( sin2x + \(\sqrt{3}cos2x\))² - 5 = cos \(\left(2x-\dfrac{\pi}{6}\right)\)

l, \(\dfrac{2.\left(cos^6x+sin^6x\right)-sinx.cosx}{\sqrt{2}-2sinx}=0\)

m, \(\dfrac{\left(1+sinx+cos2x\right).sin\left(x+\dfrac{\pi}{4}\right)}{1+tanx}=\dfrac{1}{\sqrt{2}}cosx\)

Mọi người giúp mình nha ! Mình cần gấp cho ngày mai

Chứng minh rằng \(f'\left(x\right)=0;\forall x\in R\) nếu :

a) \(f\left(x\right)=3\left(\sin^4x+\cos^4x\right)-2\left(\sin^6x+\cos^6x\right)\)

b) \(f\left(x\right)=\cos^6x+2\sin^4x.\cos^2x+3\sin^2x\cos^4x+\sin^4x\)

c) \(f\left(x\right)=\cos\left(x-\dfrac{\pi}{3}\right)\cos\left(x+\dfrac{\pi}{4}\right)+\cos\left(x+\dfrac{\pi}{6}\right)\cos\left(x+\dfrac{3\pi}{4}\right)\)

d) \(f\left(x\right)=\cos^2x+\cos^2\left(\dfrac{2\pi}{3}+x\right)+\cos^2\left(\dfrac{2\pi}{3}-x\right)\)

Tìm những giá trị của x để giá trị của các hàm số tương ứng sau bằng nhau :

a) \(y=\cos\left(2x-\dfrac{\pi}{3}\right)\) và \(y=\cos\left(\dfrac{\pi}{4}-x\right)\)

b) \(y=\sin\left(3x-\dfrac{\pi}{4}\right)\) và \(y=\sin\left(x+\dfrac{\pi}{6}\right)\)

c) \(y=\tan\left(2x+\dfrac{\pi}{5}\right)\) và \(y=\tan\left(\dfrac{\pi}{5}-x\right)\)

d) \(y=\cot3x\)              và \(y=\cot\left(x+\dfrac{\pi}{3}\right)\)

Tính đạo hàm của các hàm số sau :

a) \(y=\dfrac{1+x-x^2}{1-x+x^2}\)

b) \(y=\dfrac{\left(2-x^2\right)\left(3-x^3\right)}{\left(1-x\right)^2}\)

c) \(y=\cos2x-2\sin x\)

d) \(y=\dfrac{\cos x}{2\sin^2x}\)

e) \(y=\cos^2\dfrac{x}{3}\tan\dfrac{x}{2}\)

f) \(y=\sqrt{\sin\left(2x-\dfrac{\pi}{6}\right)}\)

g) \(y=\cos\dfrac{x}{x+1}\)

h) \(y=\dfrac{x^2-1}{\sin3x}\)

i) \(y=3\sin^2x\cos x+\cos^2x\)

k) \(y=\sqrt{7-4x}\cot3x\)

Chứng minh các hệ thức sau :

a) \(\sin\alpha+\sin\left(\alpha+\dfrac{14}{3}\pi\right)+\sin\left(\alpha-\dfrac{8}{3}\pi\right)=0\)

b) \(\dfrac{\sin4a}{1+\cos4a}.\dfrac{\cos2a}{1+\cos2a}=\cot\left(\dfrac{3}{2}\pi-a\right)\)

c) \(\left(\cos a-\cos b\right)^2-\left(\sin a-\sin b\right)^2=-4\sin^2\dfrac{a-b}{2}\cos\left(a+b\right)\)

d) \(\sin^2\left(45^0+\alpha\right)-\sin^2\left(30^0-\alpha\right)-\sin15^0\cos\left(15^0+2\alpha\right)=\sin2\alpha\)