Rút gọn bthuc \(T=1+\left(\frac{sin\alpha+tan\alpha}{cosa+1}\right)^2\)
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trang 19 SGK Toán 11 tập 1 - Chân trời ság tạo
16 tháng 7 2023 lúc 16:56
Rút gọn các biểu thức sau:
a) \(\frac{1}{{\tan \alpha + 1}} + \frac{1}{{\cot \alpha + 1}}\)
b) \(\cos \left( {\frac{\pi }{2} - \alpha } \right) - \sin \left( {\pi + \alpha } \right)\)
c) \(\sin \left( {\alpha - \frac{\pi }{2}} \right) + \cos \left( { - \alpha + 6\pi } \right) - \tan \left( {\alpha + \pi } \right)\cot \left( {3\pi - \alpha } \right)\)
\(a,\dfrac{1}{tan\alpha+1}+\dfrac{1}{cot\alpha+1}\\ =\dfrac{cot\alpha+1+tan\alpha+1}{\left(tan\alpha+1\right)\left(cot\alpha+1\right)}\\ =\dfrac{tan\alpha+cot\alpha+2}{tan\alpha\cdot cot\alpha+tan\alpha+cot\alpha+1}\\ =\dfrac{tan\alpha+cot\alpha+2}{tan\alpha+cot\alpha+2}\\ =1\)
\(b,cos\left(\dfrac{\pi}{2}-\alpha\right)-sin\left(\pi+\alpha\right)\\ =sin\alpha+sin\alpha\\ =2sin\alpha\)
\(c,sin\left(\alpha-\dfrac{\pi}{2}\right)+cos\left(-\alpha+6\pi\right)-tan\left(\alpha+\pi\right)cot\left(3\pi-\alpha\right)\\ =-sin\left(\dfrac{\pi}{2}-\alpha\right)+cos\left(\alpha\right)-tan\left(\alpha\right)cot\left(\pi-\alpha\right)\\ =-cos\left(\alpha\right)+cos\left(\alpha\right)+tan\left(\alpha\right)\cdot cot\left(\alpha\right)\\ =1\)
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Rút gọn biểu thức:
\(B=\left(1+tan^2\alpha\right)\left(1-sin^2\alpha\right)-\left(1+cot^2\alpha\right)\left(1-cos^2\alpha\right)\)
\(B=\left(1+\dfrac{sin^2a}{cos^2a}\right).cos^2a-\left(1+\dfrac{cos^2a}{sin^2a}\right).sin^2a\)
\(=\dfrac{\left(sin^2a+cos^2a\right)}{cos^2a}.cos^2a-\left(\dfrac{sin^2a+cos^2a}{sin^2a}\right).sin^2a\)
\(=1-1=0\)
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Rút gọn biểu thức: P=\(\tan\alpha\left(\frac{1+\cos^2\alpha}{\sin\alpha}-\sin\alpha\right)\)
\(P=tana\left(\frac{1+cos^2a}{sina}-sina\right)=\frac{sina}{cosa}\left(\frac{1+cos^2a-sin^2a}{sina}\right)\)
\(=\frac{sina}{cosa}.\frac{2cos^2a}{sina}=2cosa\)
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1/Rút gọn :
\(G=\left(1-sin^2x\right)cot^2x+1-cot^2x\)
2/ Nếu \(tan\alpha+cot\alpha=2\) thì \(tan^2\alpha+cot^2\alpha\) bằng bao nhiêu?
\(G=cot^2x-sin^2x.cot^2x+1-cot^2x=1-sin^2x.cot^2x\)
\(=1-sin^2x.\dfrac{cos^2x}{sin^2x}=1-cos^2x=sin^2x\)
2.
\(tana+cota=2\Rightarrow\left(tana+cota\right)^2=4\)
\(\Rightarrow tan^2a+cot^2a+2tana.cota=4\)
\(\Rightarrow tan^2a+cot^2a+2=4\)
\(\Rightarrow tan^2a+cot^2a=2\)
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Rút gọn biểu thức: P=\(\left(\frac{\sin\alpha+\tan\alpha}{\cos\alpha+1}\right)^2+1\)
\(P=\left(\frac{sina+\frac{sina}{cosa}}{cosa+1}\right)^2+1=\left(\frac{sina\left(cosa+1\right)}{cosa\left(cosa+1\right)}\right)^2+1\)
\(=\left(\frac{sina}{cosa}\right)^2+1=\frac{sin^2a+cos^2a}{cos^2a}=\frac{1}{cos^2a}\)
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trang 23 SGK Toán 11 tập 1 - Chân trời sáng tạo
16 tháng 7 2023 lúc 17:04
Rút gọn các biểu thức sau:
a, \(\sqrt 2 \sin \left( {\alpha + \frac{\pi }{4}} \right) - cos\alpha \),
b, \({\left( {cos\alpha + \sin \alpha } \right)^2} - \sin 2\alpha \)
\(a,\sqrt{2}sin\left(\alpha+\dfrac{\pi}{4}\right)-cos\alpha\\ =\sqrt{2}\left(sin\alpha cos\dfrac{\pi}{4}+cos\alpha sin\dfrac{\pi}{4}\right)-cos\alpha\\ =\sqrt{2}\left(sin\alpha\cdot\dfrac{\sqrt{2}}{2}+cos\alpha\cdot\dfrac{\sqrt{2}}{2}\right)-cos\alpha\\ =\sqrt{2}\cdot sin\alpha\cdot\dfrac{\sqrt{2}}{2}+\sqrt{2}\cdot cos\alpha\cdot\dfrac{\sqrt{2}}{2}-cos\alpha\\ =sin\alpha+cos\alpha-cos\alpha\\ =sin\alpha\)
\(b,\left(cos\alpha+sin\alpha\right)^2-sin2\alpha\\ =cos^2\alpha+sin^2\alpha=2cos\alpha sin\alpha-2sin\alpha cos\alpha\\ =sin^2\alpha+cos^2\alpha\\ =1\)
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2) Rút gọna)1-sin^22b)left(1-cosalpharight)left(1+cosalpharight)c)1+sin^2alpha+cos^2alphad)sinalpha-sinalpha.cos^2alphae)sin^2alpha+cos^2alpha+2sin^2alpha.cos^2alphaf)tan^2alpha-sin^2alpha.tan^2alphag)cos^2alpha+tan^2alpha.cos^2alphah)tan^2alphaleft(2cos^2alpha+sin^2alpha-1right)
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2) Rút gọn
a)\(1-\sin^22\)
b)\(\left(1-\cos\alpha\right)\left(1+\cos\alpha\right)\)
c)\(1+\sin^2\alpha+\cos^2\alpha\)
d)\(\sin\alpha-\sin\alpha.\cos^2\alpha\)
e)\(\sin^2\alpha+\cos^2\alpha+2\sin^2\alpha.\cos^2\alpha\)
f)\(\tan^2\alpha-\sin^2\alpha.\tan^2\alpha\)
g)\(\cos^2\alpha+\tan^2\alpha.\cos^2\alpha\)
h)\(\tan^2\alpha\left(2\cos^2\alpha+\sin^2\alpha-1\right)\)
\(a,1-sin^2\alpha=cos^2\alpha\)
\(b,\left(1-cos\alpha\right)\left(1+cos\alpha\right)=1-cos^2\alpha=sin^2\alpha\)
\(c,1+sin^2\alpha+cos^2\alpha=1+1=2\)
\(d,sin\alpha-sin\alpha.cos^2\alpha=sin\alpha.\left(1-cos^2\alpha\right)=sin\alpha.sin^2\alpha=sin^3\alpha\)
\(e,sin^2\alpha+cos^2\alpha+2sin^2\alpha.cos^2\alpha\)
\(=1+2sin^2\alpha.cos^2\alpha\)
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CMR
a)\(\frac{1+\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1-\cos\alpha}\)
b)\(\frac{\tan\alpha+1}{\tan\alpha-1}=\frac{1+\cot\alpha}{1-\cot\alpha}\)
c) \(\tan^2\alpha-\sin^2\alpha=\tan^2\alpha.\sin^2\alpha\)
d)\(\frac{1-4\sin^2\alpha.\cos^2\alpha}{\left(\sin\alpha-\cos\alpha\right)^2}=\left(\sin\alpha+\cos\alpha\right)^2\)
H= \(\sin^4\alpha\left(1+2\cos^2\alpha\right)+\cos^4\alpha\left(1+2\sin^2\alpha\right)\)
K=\(\tan^2\alpha\left(2\cos^2\alpha+\sin^2\alpha-1\right)\)
Rút gọn biểu thức