Chứng minh: \(tan^2x-sin^2x=tan^2x.sin^2x\)
Những câu hỏi liên quan
CM đẳng thức: \(\frac{tan^2x-tan^2y}{tan^2x.tan^2y}=\frac{sin^2x-sin^2y}{sin^2x.sin^2y}\)
Cho \(0< x< 90^o\) . Chứng minh giá trị biểu thức sau không phụ thuộc vào giá trị của biến:
\(A=sin^6x+cos^6x+3sin^2x.cos^2x+tan^2x.cos^2x+cot^2x.sin^2x\)
Chứng minh: 1.dfrac{cot^2x-sin^2x}{cot^2x-tan^2x}sin^2xcdotcos^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}cdotdfrac{cot^2x-1}{cot x}1 5.dfrac{1+sin^2x}{1-sin^2x}1+2tan^2x
Đọc tiếp
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.\(\dfrac{1+\sin^2x}{1-\sin^2x}=1+2\tan^2x\)
Câu 1 đề sai, chắc chắn 1 trong 2 cái \(cot^2x\) phải có 1 cái là \(cos^2x\)
2.
\(\dfrac{1-sinx}{cosx}-\dfrac{cosx}{1+sinx}=\dfrac{\left(1-sinx\right)\left(1+sinx\right)-cos^2x}{cosx\left(1+sinx\right)}=\dfrac{1-sin^2x-cos^2x}{cosx\left(1+sinx\right)}\)
\(=\dfrac{1-\left(sin^2x+cos^2x\right)}{cosx\left(1+sinx\right)}=\dfrac{1-1}{cosx\left(1+sinx\right)}=0\)
3.
\(\dfrac{tanx}{sinx}-\dfrac{sinx}{cotx}=\dfrac{tanx.cotx-sin^2x}{sinx.cotx}=\dfrac{1-sin^2x}{sinx.\dfrac{cosx}{sinx}}=\dfrac{cos^2x}{cosx}=cosx\)
4.
\(\dfrac{tanx}{1-tan^2x}.\dfrac{cot^2x-1}{cotx}=\dfrac{tanx}{1-tan^2x}.\dfrac{\dfrac{1}{tan^2x}-1}{\dfrac{1}{tanx}}=\dfrac{tanx}{1-tan^2x}.\dfrac{1-tan^2x}{tanx}=1\)
5.
\(\dfrac{1+sin^2x}{1-sin^2x}=\dfrac{1+sin^2x}{cos^2x}=\dfrac{1}{cos^2x}+tan^2x=\dfrac{sin^2x+cos^2x}{cos^2x}+tan^2x\)
\(=tan^2x+1+tan^2x=1+2tan^2x\)
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chứng minh (tan^2x-sin^2x)/(cot^2x-cos^2x)=tan^6x
\(VT=\frac{\frac{\sin^2x}{\cos^2x}-\sin^2x}{\frac{\cos^2x}{\sin^2x}-\cos^2x}=\frac{\frac{\sin^2x-\sin^2x.\cos^2x}{\cos^2x}}{\frac{\cos^2x-\cos^2x.\sin^2x}{\sin^2x}}\)
\(=\frac{\sin^2x}{\cos^2x}.\frac{\sin^2x-\sin^2x.\cos^2x}{\cos^2-\cos^2x.\sin^2x}\)
\(=\frac{\sin^2x}{\cos^2x}.\frac{\tan^2x-\sin^2x}{\cos^2x}=\frac{\sin^2x}{\cos^2x}.\left(\frac{\tan^2x}{\cos^2x}-\tan^2x\right)\)
\(1+\tan^2x=\frac{1}{\cos^2x}\Rightarrow\frac{\tan^2x}{\cos^2x}=\tan^2x\left(1+\tan^2x\right)\)
\(\Rightarrow VT=\tan^2x.\tan^4x=\tan^6x=VP\)
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\(\frac{tan^2x-sin^2x}{cot^2x-cos^2x}=\frac{sin^2x.cos^2x\left(tan^2x-sin^2x\right)}{sin^2x.cos^2x\left(cot^2x-cos^2x\right)}=\frac{sin^4x\left(1-cos^2x\right)}{cos^4x\left(1-sin^2x\right)}=\frac{sin^6x}{cos^6x}=tan^6x\)
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Chứng minh đẳng thức: \(\dfrac{sin^2x-cos^2x+cos^4x}{cos^2x-sin^2x+sin^4x}=tan^4x\)
\(\dfrac{sin^2x-cos^2x+cos^4x}{cos^2x-sin^2x+sin^4x}=\dfrac{1-2cos^2x+cos^4x}{1-2sin^2x+sin^4x}==\dfrac{\left(cos^2x-1\right)^2}{\left(sin^2-1\right)^2}=\dfrac{sin^4x}{cos^4x}=tan^4x\)
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chứng minh rằng
a) tanx(cot\(^2\)x - 1) = cotx(1 - tan\(^2\)x)
b) tan\(^2\)x - sin\(^2\)x = tan\(^2\)x.sin\(^2\)x
c) \(\dfrac{cos^2x-sin^2x}{cot^2x-tan^2x}\) - cos\(^2\)x = - cos\(^4\)x
a: tan x(cot^2x-1)
\(=\dfrac{1}{cotx}\left(cot^2x-cotx\cdot tanx\right)\)
=cotx-tanx/cotx=cotx(1-tan^2x)
b: \(tan^2x-sin^2x=\dfrac{sin^2x}{cos^2x}-sin^2x\)
\(=sin^2x\left(\dfrac{1}{cos^2x}-1\right)=sin^2x\cdot\dfrac{sin^2x}{cos^2x}=sin^2x\cdot tan^2x\)
c: \(\dfrac{cos^2x-sin^2x}{cot^2x-tan^2x}=\dfrac{cos^2x-sin^2x}{\dfrac{cos^2x}{sin^2x}-\dfrac{sin^2x}{cos^2x}}\)
\(=\left(cos^2x-sin^2x\right):\dfrac{cos^4x-sin^4x}{sin^2x\cdot cos^2x}\)
\(=\dfrac{sin^2x\cdot cos^2x}{1}=sin^2x\cdot cos^2x\)
=>sin^2x*cos^2x-cos^2x=cos^2x(sin^2x-1)
=-cos^2x*cos^2x=-cos^4x
=>ĐPCM
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chứng minh 1-tan^2x/1+tan^2x=cos^4x-sin^4x
\(\frac{1-tan^2x}{1+tan^2x}=\frac{cos^2x\left(1-tan^2x\right)}{cos^2x\left(1+tan^2x\right)}=\frac{cos^2x-sin^2x}{cos^2x+sin^2x}=cos^2x-sin^2x\)
\(=\left(cos^2x-sin^2x\right).1=\left(cos^2x-sin^2x\right)\left(cos^2x+sin^2x\right)=cos^4x-sin^4x\)
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Chứng minh rằng: \(\dfrac{-cos^2x.sin^2x+mcosx-3m+1}{sin^2x-cosx-3}=m\) luôn có nghiệm với mọi m > 1.
Chứng minh phương trình \(\dfrac{-cos^2x.sin^2x+mcosx-3m+1}{sin^2x-cosx-3}=m\) luôn có nghiệm với mọi m > 1