Chứng minh:
1.\(\dfrac{\cot^2x-\sin^2x}{\cot^2x-\tan^2x}=\sin^2x\cdot\cos^2x\)
2.\(\dfrac{1-\sin x}{\cos x}-\dfrac{\cos x}{1+\sin x}=0\)
3.\(\dfrac{\tan x}{\sin x}-\dfrac{\sin x}{\cot x}=\cos x\)
4.\(\dfrac{\tan x}{1-\tan^2x}\cdot\dfrac{\cot^2x-1}{\cot x}=1\)
5.\(\dfrac{1+\sin^2x}{1-\sin^2x}=1+2\tan^2x\)
chứng minh các đẳng thức sau : a) \(\frac{1+2sinxcosx}{sin^2x-cos^2x}\) = \(\frac{tan+1}{tan-1}\) ; b) sin4x - cos4x = 1 - 2cos2x ; c) sin4x + cos4x = \(\frac{3}{4}\) + \(\frac{1}{4}\)cosx ; d) sin6x + cos6x = \(\frac{5}{8}\) + \(\frac{3}{8}\)cos4x ; e) cotx - tanx = 2cot2x ; f) \(\frac{sin2x+sin4x+sin6x}{1+cos2x+cos4x}\) = 2sin2x
Chứng minh các biểu thức sau không phụ thuộc vào x:
1, \(A=3\left(sin^4x+cos^4x\right)-2\left(sin^6x+cos^6x\right)\)
2, \(B=cos^6x+2sin^4x.cos^2x+3sin^2x.cos^4x+sin^4x\)
3, \(C=cos\left(x-\dfrac{\pi}{3}\right).cos\left(x+\dfrac{\pi}{4}\right)+cos\left(x+\dfrac{\pi}{6}\right).cos\left(x+\dfrac{3\pi}{4}\right)\)
4, \(D=cos^2x+cos^2\left(x+\dfrac{2\pi}{3}\right)+cos^2\left(\dfrac{2\pi}{3}-x\right)\)
5, \(E=2\left(sin^4x+cos^4x+sin^2x.cos^2x\right)-\left(sin^8x+cos^8x\right)\)
6, \(F=cos\left(\pi-x\right)+sin\left(\dfrac{-3\pi}{2}+x\right)-tan\left(\dfrac{\pi}{2}+x\right).cot\left(\dfrac{3\pi}{2}-x\right)\)
Chứng minh đẳng thức:
1 ,\(tan\left(\dfrac{\pi}{4}+\dfrac{x}{2}\right)+cot\left(\dfrac{\pi}{4}+\dfrac{x}{2}\right)=\dfrac{2}{cosx}\)
2 ,\(sin^8x-cos^8x=-\left(\dfrac{7}{8}cos2x+\dfrac{1}{8}cos6x\right)\)
3 ,\(3-4cos2x+cos4x=8sin^4x\)
4 ,\(sin\left(2x+\dfrac{\pi}{3}\right).cos\left(x-\dfrac{\pi}{6}\right)-cos\left(2x+\dfrac{\pi}{3}\right).cos\left(\dfrac{2\pi}{3}-x\right)=cosx\)
5 ,\(\sqrt{3}cos2x+sin2x+sin\left(4x-\dfrac{\pi}{3}\right)=4cos\left(2x-\dfrac{\pi}{6}\right).sin^2\left(x+\dfrac{\pi}{6}\right)\)
Chứng minh đẳng thức: \(\dfrac{sin^2x-cos^2x+cos^4x}{cos^2x-sin^2x+sin^4x}=tan^4x\)
Cho góc x với cos=-1/2. Tính giá trị biểu thức B=cos^2x+sin^2x+tan^2x
Chứng minh
a) \(\dfrac{1+\cos x+\cos2x+\cos3x}{2\cos^2x+\cos x-1}=2\cos x\)
b) \(\cos\dfrac{5x}{2}.\cos\dfrac{3x}{2}+\sin\dfrac{7x}{2}.\sin\dfrac{x}{2}=\cos x.\cos2x\)
cho tam giác ABC , chứng minh rằng : a) sin(B + C) = sinA ; b) cos(A + B) = -cosC ; c) sin\(\frac{B+C}{2}\) = cos\(\frac{A}{2}\) ; d) tan\(\frac{A+C}{2}\) = cot\(\frac{B}{2}\)
Chứng minh đẳng thức :
a) \(\dfrac{\cos\left(a-b\right)}{\cos\left(a+b\right)}=\dfrac{\cot a.\cot b+1}{\cot a.\cot b-1}\)
b) \(\sin\left(a+b\right)\sin\left(a-b\right)=\sin^2a-\sin^2b=\cos^2b-\cos^2a\)
c) \(\cos\left(a+b\right)\cos\left(a-b\right)=\cos^2a-\sin^2b=\cos^2b-\sin^2a\)