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TK
1
25 tháng 10 2023
a: \(\left(1-cosx\right)\left(1+cosx\right)=1^2-cos^2x=sin^2x\)
b: \(tan^2x\left(2cos^2x+sin^2x-1\right)\)
\(=tan^2x\left(1-1+cos^2x\right)\)
\(=\dfrac{sin^2x}{cos^2x}\cdot cos^2x=sin^2x\)
c: \(sin^4x+cos^4x+2\cdot cos^2x\cdot sin^2x\)
\(=\left(sin^2x+cos^2x\right)^2\)
\(=1^2=1\)
5 tháng 7 2019
\(A=\left(\sin\alpha+\cos\alpha+\sin\alpha-\cos\alpha\right)^2-2\left(\sin\alpha+\cos\alpha\right)\left(\sin\alpha-\cos\alpha\right)\)
\(=4\sin^2\alpha-2\sin^2\alpha+2\cos^2\alpha=2\left(\sin^2\alpha+\cos^2\alpha\right)=2\)
\(B=\sin^4\alpha+\cos^4\alpha+2\sin^2\alpha.\cos^2\alpha\left(\sin^2\alpha+\cos^2\alpha\right)=\sin^4\alpha+\cos^4\alpha+2\sin^2\alpha.\cos^2\alpha\)
\(=\left(\sin^2\alpha+\cos^2\alpha\right)^2-1=0\)
\(C=3\left(\sin^4\alpha+\cos^4\alpha\right)-2\sin^2\alpha.\cos^2\alpha\left(\sin^2\alpha+\cos^2\alpha\right)=3\left(\sin^4\alpha+\cos^4\alpha\right)-2\sin^2\alpha.\cos^2\alpha\)
\(=3\left(\sin^2\alpha+\cos^2\alpha-\frac{1}{9}\right)^2-\frac{1}{9}=\frac{61}{27}\)
29 tháng 10 2018
a) 1- \(sin^2\alpha\)= \(cos^2\alpha\)
b) (\(1-cos\alpha\))(\(1+cos\alpha\)) = 1 - cos2\(\alpha\) = sin2\(\alpha\)
c) 1 + cos2\(\alpha\) + sin2\(\alpha\) = \(1+1=2\)
d) sin\(\alpha\) - sin\(\alpha.cos^2\alpha\)
= \(sin\alpha\left(1-cos^2\alpha\right)=sin\alpha.sin^2\alpha=sin^3\alpha\)
e) \(sin^4\alpha+cos^4\alpha+2sin^2\alpha.cos^2\alpha\)
= \(\left(sin^2\alpha\right)^2+2sin^2\alpha.cos^2\alpha+\left(cos^2\alpha\right)^2\)
= \(\left(sin^2\alpha+cos^2\alpha\right)^2=1^2=1\)
f) \(tan^2\alpha-sin^2\alpha.tan^2\alpha\)
= \(tan^2\alpha\left(1-sin^2\alpha\right)=tan^2\alpha.cos^2\alpha=sin^2\alpha\)
g) \(cos^2\alpha+tan^2\alpha.cos^2\alpha\)
= \(cos^2\alpha\left(1+tan^2\alpha\right)=cos^2\alpha.\dfrac{1}{cos^2\alpha}=1\)
h) \(tan^2\alpha\left(2cos^2\alpha+sin^2\alpha-1\right)\)
= \(tan^2\alpha\left[cos^2\alpha+\left(cos^2\alpha+sin^2\alpha\right)-1\right]\)
= \(tan^2\alpha\left(cos^2\alpha+1-1\right)\)
= \(tan^2\alpha.cos^2\alpha=sin^2\alpha\)
NV
Nguyễn Việt Lâm
Giáo viên
1 tháng 9 2020
\(A=\frac{sina+cosa}{cosa-sina}=\frac{\frac{sina}{cosa}+\frac{cosa}{cosa}}{\frac{cosa}{cosa}-\frac{sina}{cosa}}=\frac{tana+1}{1-tana}=\frac{5+1}{1-5}=...\)
\(B=\frac{8cos^3a-2sin^3a+cosa}{2cosa-sin^3a}\) để làm được câu này chỉ cần nhớ đến công thức: \(\frac{1}{cos^2a}=1+tan^2a\)
\(B=\frac{\frac{8cos^3a}{cos^3a}-\frac{2sin^3a}{cos^3a}+\frac{cosa}{cosa}.\frac{1}{cos^2a}}{\frac{2cosa}{cosa}.\frac{1}{cos^2a}-\frac{sin^3a}{cos^3a}}=\frac{8-2tan^3a+1+tan^2a}{2\left(1+tan^2a\right)-tan^3a}=\frac{9-2tan^3a+tan^2a}{2+2tan^2a-tan^3a}=\frac{9-2.5^3+5^2}{2+2.5^2-5^3}=...\)
Y
15 tháng 7 2019
\(\left(1-\cos\alpha\right)\left(1+\cos\alpha\right)=1-\cos^2\alpha=\left(\sin^2\alpha+\cos^2\alpha\right)-\cos^2\alpha\\
=\sin^2\alpha\)
\(1+\sin^2\alpha+\cos^2\alpha=1+1=2\)
\(\sin^4\alpha+\cos^4\alpha+2\sin^2\alpha\cdot\cos^2\alpha\\ =\left(\sin^2\alpha\right)^2+2\sin^2\alpha\cdot\cos^2\alpha+\left(\cos^2\alpha\right)^2\\ =\left(\sin^2\alpha+\cos^2\alpha\right)^2\\ =1^2=1\)
Y
15 tháng 7 2019
\(\tan^2\alpha-\sin^2\alpha\cdot\tan^2\alpha\\ =\tan^2\alpha\left(1-\sin^2\alpha\right)\\ =\left(\frac{\sin\alpha}{\cos\alpha}\right)^2\cdot\cos^2\alpha\\ =\frac{\sin^2\alpha}{\cos^2\alpha}\cdot\cos^2\alpha\\ =\sin^2\alpha\)
\(\cos^2\alpha+\tan^2\alpha\cdot\cos^2\alpha\\ =\cos^2\alpha+\left(\frac{\sin\alpha}{\cos\alpha}\right)^2\cdot\cos^2\alpha\\ =\cos^2\alpha+\frac{\sin^2\alpha}{\cos^2\alpha}\cdot\cos^2\alpha\\ =\cos^2\alpha+\sin^2\alpha\\ =1\)
\(\tan^2\alpha\cdot\left(2\cos^2\alpha+\sin^2\alpha-1\right)\\ =\tan^2\alpha\cdot\left(2\cos^2\alpha+\sin^2\alpha-\sin^2\alpha-\cos^2\alpha\right)\\ =\tan^2\alpha\cdot\cos^2\alpha\\ =\frac{\sin^2\alpha}{\cos^2\alpha}\cdot\cos^2\alpha=\sin^2\alpha\)
16 tháng 8 2023
\(B=tan^267^0-cot^223^0+2\cdot\left(sin^216^0+cos^216^0\right)-2\)
\(=0+2\cdot1-2=0\)
\(A=cot67\cdot tan67-2\left(\dfrac{\sqrt{2}}{2}\cdot sin64\right)^2-2\cdot\dfrac{sin23}{3\cdot sin23}-sin^226^0\)
\(=1-2\cdot\dfrac{1}{2}\cdot sin^264^0-\dfrac{2}{3}-sin^226^0\)
\(=1-1-\dfrac{2}{3}=-\dfrac{2}{3}\)
TN
1
\(=sin^4x+cos^4x+2sin^4x\cdot cos^2x+2\cdot sin^2x\cdot cos^4x\)
\(=sin^4x+cos^4x+2\cdot sin^2x\cdot cos^2x\)
\(=\left(sin^2x+cos^2x\right)^2=1\)