Chứng ming: \(\cot^2\alpha-\cos^2\alpha=\cot^2\cdot\cos^2\alpha\)
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Chứng minh:
a)cot^2alpha-cos^2alphacdot cot^2alphacos^2alpha
b)tan^2alpha-sin^2alphacdot tan^2alphasin^2alpha
c) dfrac{1-cos^2}{sinalpha} dfrac{sinalpha}{1+cosalpha}
d)tan^2alpha-sin^2alphatan^2cdot sin^2alpha
e) sin^6alpha+cos^6alpha+3sin^2cdot cos^2alpha1
Đọc tiếp
Chứng minh:
a)\(cot^2\alpha-cos^2\alpha\cdot cot^2\alpha=cos^2\alpha\)
b)\(tan^2\alpha-sin^2\alpha\cdot tan^2\alpha=sin^2\alpha\)
c) \(\dfrac{1-cos^2}{sin\alpha}\) = \(\dfrac{sin\alpha}{1+cos\alpha}\)
d)\(tan^2\alpha-sin^2\alpha=tan^2\cdot sin^2\alpha\)
e) \(\sin^6\alpha+cos^6\alpha+3sin^2\cdot cos^2\alpha=1\)
27 tháng 10 2017 lúc 17:11
Chứng minh giá trị các biểu thức sau luôn là hằng số với mọi góc nhọn \(\alpha\)
\(a.\sin^4\alpha+\cos^4\alpha+2\sin^2\alpha\cdot\cos^2\alpha\)
\(b.\cos^2\alpha+\sin^2\alpha+\tan^2\alpha\cdot\cos^2\alpha+\cot^2\alpha\cdot\sin^2\alpha\)
CMR:\(1,\tan\alpha\cdot\cot\alpha=1\)
\(2,\sin^2\alpha+\cos^2\alpha=1\)
\(3,\tan\alpha=\dfrac{\sin\alpha}{\cos\alpha};\cot\alpha=\dfrac{\cos\alpha}{\tan\alpha}\)
Chứng minh các đẳng thức sau:
a, sin^4alpha-cos^4alpha+12sin^2alpha
b,dfrac{sin^2alpha+2cos^2alpha-1}{cot^2alpha}sin^2alpha
c, dfrac{1-sin^2alpha.cos^2alpha}{cos^2alpha}-cos^2alphatan^2alpha
d, dfrac{sin^2alpha-tan^2alpha}{cos^2alpha-cot^2alpha}tan^6alpha
e, left(1+cotalpharight)sin^3alpha+left(1+tanalpharight)cos^3alphasinalpha.cosalpha
f,dfrac{left(sinalpha+cosalpharight)^2-1}{cotalpha-sinalpha.cosalpha}2tan^2alpha
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Chứng minh các đẳng thức sau:
a, \(\sin^4\alpha-\cos^4\alpha+1=2\sin^2\alpha\)
b,\(\dfrac{\sin^2\alpha+2\cos^2\alpha-1}{\cot^2\alpha}=\sin^2\alpha\)
c, \(\dfrac{1-\sin^2\alpha.\cos^2\alpha}{\cos^2\alpha}-\cos^2\alpha=\tan^2\alpha\)
d, \(\dfrac{\sin^2\alpha-\tan^2\alpha}{\cos^2\alpha-\cot^2\alpha}=\tan^6\alpha\)
e, \(\left(1+\cot\alpha\right)\sin^3\alpha+\left(1+\tan\alpha\right)\cos^3\alpha=\sin\alpha.\cos\alpha\)
f,\(\dfrac{\left(\sin\alpha+\cos\alpha\right)^2-1}{\cot\alpha-\sin\alpha.\cos\alpha}=2\tan^2\alpha\)
a)
\(\sin ^4a-\cos ^4a+1=(\sin ^2a-\cos ^2a)(\sin ^2a+\cos^2a)+1\)
\(=(\sin ^2a-\cos ^2a).1+1=\sin ^2a-\cos ^2a+\sin ^2a+\cos ^2a\)
\(=2\sin ^2a\)
b) \(\sin ^2a+2\cos ^2a-1=(\sin ^2a+\cos^2a)+\cos ^2a-1\)
\(=1+\cos ^2a-1=\cos ^2a\)
\(\Rightarrow \frac{\sin ^2a+2\cos ^2a-1}{\cot ^2a}=\frac{\cos ^2a}{\cot ^2a}=\frac{\cos ^2a}{\frac{\cos ^2a}{\sin ^2a}}=\sin ^2a\)
c)
\(\frac{1-\sin ^2a\cos ^2a}{\cos ^2a}-\cos ^2a=\frac{1}{\cos ^2a}-\sin ^2a-\cos ^2a\)
\(=\frac{1}{\cos ^2a}-(\sin ^2a+\cos ^2a)=\frac{1}{\cos ^2a}-1\)
\(=\frac{1-\cos ^2a}{\cos ^2a}=\frac{\sin ^2a}{\cos ^2a}=\tan ^2a\)
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d)
\(\frac{\sin ^2a-\tan ^2a}{\cos ^2a-\cot ^2a}=\frac{\sin ^2a-\frac{\sin ^2a}{\cos ^2a}}{\cos ^2a-\frac{\cos ^2a}{\sin ^2a}}\) \(=\frac{\sin ^2a(1-\frac{1}{\cos ^2a})}{\cos ^2a(1-\frac{1}{\sin ^2a})}\)
\(=\frac{\sin ^2a.\frac{\cos ^2a-1}{\cos ^2a}}{\cos ^2a.\frac{\sin ^2a-1}{\sin ^2a}}\) \(=\frac{\sin ^2a.\frac{-\sin ^2a}{\cos ^2a}}{\cos ^2a.\frac{-\cos ^2a}{\sin ^2a}}=\frac{\sin ^6a}{\cos ^6a}=\tan ^6a\)
f)
\(\frac{(\sin a+\cos a)^2-1}{\cot a-\sin a\cos a}=\frac{\sin ^2a+\cos ^2a+2\sin a\cos a-1}{\frac{\cos a}{\sin a}-\sin a\cos a}\)
\(=\sin a.\frac{1+2\sin a\cos a-1}{\cos a-\cos a\sin ^2a}\)
\(=\sin a. \frac{2\sin a\cos a}{\cos a(1-\sin ^2a)}=\sin a. \frac{2\sin a\cos a}{\cos a. \cos^2 a}=\frac{2\sin ^2a}{\cos ^2a}=2\tan ^2a\)
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e)
\((1+\cot a)\sin ^3a+(1+\tan a)\cos ^3a\)
\(=(\sin ^3a+\cos ^3a)+\cot a.\sin ^3a+\tan a.\cos^3a\)
\(=(\sin a+\cos a)(\sin ^2a-\sin a\cos a+\cos ^2a)+\frac{\cos a}{\sin a}.\sin ^3a+\frac{\sin a}{\cos a}.\cos ^3a\)
\(=(\sin a+\cos a)(1-\sin a\cos a)+\cos a\sin ^2a+\sin a\cos ^2a\)
\(=\sin a+\cos a-\sin a\cos a(\sin a+\cos a)+\cos a\sin a(\sin a+\cos a)\)
\(=\sin a+\cos a\)
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Chứng minh các hệ thức sau:
\(\cot^2\alpha-\cos^2\alpha=\cot^2\alpha.\cos^2\alpha\)
Ta có \(VT=\cot^2a-\cos^2a=\frac{\cos^2a}{\sin^2a}-\cos^2a=\frac{\cos^2a-\cos^2a.\sin^2a}{\sin^2a}\)
\(=\frac{\cos^2a\left(1-\sin^2a\right)}{\sin^2a}=\frac{\cos^2a.\cos^2a}{\sin^2a}=\cot^2a.\cos^2a=VP\left(đpcm\right)\)
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chứng minh đẳng thức
cot2 α - cos2 α = cot2α.cos2α
\(1+\cot^2\alpha=\frac{1}{\sin^2\alpha}\)
\(\Rightarrow VT=\frac{1}{\sin^2\alpha}-1-\left(1-\sin^2\alpha\right)\)
\(=\frac{1}{\sin^2\alpha}-2+\sin^2\alpha\)
\(=\frac{1-2\sin^2\alpha+\sin^4\alpha}{\sin^2\alpha}=\frac{\left(1-\sin^2\alpha\right)}{\sin^2\alpha}=\frac{\cos^2\alpha}{\sin^2\alpha}=\cot^2\alpha\)
Ủa đề bài sai hử bạn?
Chứng minh rằng giá trị của biểu thức sau không phụ thuộc vào giá trị của góc nhọn alpha a) A frac{cot^2alpha-cos^2alpha}{cot^2alpha}-frac{sinalpha.cosalpha}{cotalpha}b) B left(cosalpha-sinalpharight)^2+left(cosalpha+sinalpharight)^2+cos^4alpha-sin^4alpha-2cos^2alphac) C sin^6x+cos^6x+3sin^2x.cos^2x
Đọc tiếp
Chứng minh rằng giá trị của biểu thức sau không phụ thuộc vào giá trị của góc nhọn \(\alpha\)
a) A = \(\frac{\cot^2\alpha-\cos^2\alpha}{\cot^2\alpha}-\frac{\sin\alpha.\cos\alpha}{\cot\alpha}\)
b) B = \(\left(\cos\alpha-\sin\alpha\right)^2+\left(\cos\alpha+\sin\alpha\right)^2+\cos^4\alpha-\sin^4\alpha-2\cos^2\alpha\)
c) C = \(\sin^6x+\cos^6x+3\sin^2x.\cos^2x\)
a/ \(A=\frac{cot^2a-cos^2a}{cot^2a}-\frac{sina.cosa}{cota}\)
\(=\frac{\frac{cos^2a}{sin^2a}-cos^2a}{\frac{cos^2a}{sin^2a}}-\frac{sina.cosa}{\frac{cosa}{sina}}\)
\(=\left(1-sin^2a\right)-sin^2a=1\)
b/ \(B=\left(cosa-sina\right)^2+\left(cosa+sina\right)^2+cos^4a-sin^4a-2cos^2a\)
\(=cos^2a-2cosa.sina+sin^2a+cos^2a+2cosa.sina+sin^2a+\left(cos^2a+sin^2a\right)\left(cos^2a-sin^2a\right)-2cos^2a\)
\(=2+\left(cos^2a-sin^2a\right)-2cos^2a\)
\(=2-sin^2a-cos^2a=2-1=1\)
c/ \(C=sin^6x+cos^6x+3sin^2x.cos^2x\)
\(=\left(sin^2x+cos^2x\right)\left(sin^4x-sin^2x.cos^2x+cos^4x\right)+3sin^2x.cos^2x\)
\(=sin^4x-sin^2x.cos^2x+cos^4x+3sin^2x.cos^2x\)
\(=sin^4x+cos^4x+2sin^2x.cos^2x\)
\(=\left(sin^2x+cos^2x\right)^2=1\)
1. Đơn giản biểu thức
a. \(\sin\alpha\cdot\cos\alpha\left(\tan\alpha+\cot\alpha\right)\)
b. \(\left(\sin^2\alpha+\cos^2\alpha\right)^2+\left(\sin\alpha-\cos\alpha\right)^2\)
c. \(\tan^2\alpha-\sin^2\alpha\cdot\tan^2\alpha\)
a) ta có : \(sin\alpha.cos\alpha\left(tan\alpha+cot\alpha\right)=sin\alpha.cos\alpha\left(\dfrac{sin\alpha}{cos\alpha}+\dfrac{cos\alpha}{sin\alpha}\right)\)
\(=sin^2\alpha+cos^2\alpha=1\)
b) ta có : \(\left(sin^2\alpha+cos^2\alpha\right)^2+\left(sin\alpha-cos\alpha\right)^2\)
\(=1^2+1-2sin\alpha.cos=2\left(1-2sin\alpha.cos\alpha\right)\)
c) ta có : \(tan^2\alpha-sin^2\alpha.tan^2\alpha=tan^2\alpha\left(1-sin^2\alpha\right)\)
\(=\dfrac{sin^2\alpha}{cos^2\alpha}.cos^2\alpha=sin^2\alpha\)
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Chứng minh các hệ thức sau:
a) 1 + tan2 α = \(\frac{1}{cos^2\alpha}\)
b) 1 + cot2 α = \(\frac{1}{sin^2\alpha}\)
c) cot2 α - cos2 α= cot2 α . cos2 α
d) \(\frac{1+cos^2\alpha}{sin\alpha}\)=\(\frac{Sin\alpha}{1-cos\alpha}\)
\(1+tan^2a=1+\frac{sin^2a}{cos^2a}=\frac{cos^2a+sin^2a}{cos^2a}=\frac{1}{cos^2a}\)
\(1+cot^2a=1+\frac{cos^2a}{sin^2a}=\frac{sin^2a+cos^2a}{sin^2a}=\frac{1}{sin^2a}\)
\(cot^2a-cos^2a=\frac{cos^2a}{sin^2a}-cos^2a=cos^2a\left(\frac{1}{sin^2a}-1\right)=cos^2a\left(\frac{1-sin^2a}{sin^2a}\right)\)
\(=cos^2a.\frac{cos^2a}{sin^2a}=cos^2a.cot^2a\)
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