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a: \(\left(sinx+cosx\right)^2=m^2\)
=>\(m^2=sin^2x+cos^2x+2\cdot sinx\cdot cosx\)
=>\(2\cdot sinx\cdot cosx=m^2-1\)
\(\left(sinx-cosx\right)^2=sin^2x+cos^2x-2\cdot sinx\cdot cosx\)
\(=1-\left(m^2-1\right)=2-m^2\)
\(\left|sin^4x-cos^4x\right|=\left|\left(sin^2x+cos^2x\right)\left(sin^2x-cos^2x\right)\right|\)
\(=\left|sin^2x-cos^2x\right|\)
\(=\left|\left(sinx+cosx\right)\left(sinx-cosx\right)\right|\)
\(=\left|m\left(2-m^2\right)\right|=\left|2m-m^3\right|\)
b: \(m=sinx+cosx\)
\(=\sqrt{2}\cdot\left(sinx\cdot\dfrac{\sqrt{2}}{2}+cosx\cdot\dfrac{\sqrt{2}}{2}\right)\)
\(=\sqrt{2}\cdot sin\left(x+\dfrac{\Omega}{4}\right)\)
=>\(\left|m\right|=\sqrt{2}\cdot\left|sin\left(x+\dfrac{\Omega}{4}\right)\right|\)
\(0< =\left|sin\left(x+\dfrac{\Omega}{4}\right)\right|< =1\)
=>\(0< =\sqrt{2}\cdot\left|sin\left(x+\dfrac{\Omega}{4}\right)\right|< =\sqrt{2}\)
=>\(\left|m\right|< =\sqrt{2}\)
\(A=\left|\sin^4x-\cos^4x\right|=\left|\left(\sin^2x\right)^2-\left(\cos^2x\right)^2\right|\)
\(A=\left|\left(1-\cos^2x\right)^2-\left(\cos^2x\right)^2\right|=\left|1-2\cos^2x+\cos^4x-\cos^4x\right|\)
\(=\left|1-2\cos^2x\right|=\left|\sin^2x-\cos^2x\right|=\left|\left(\sin x-\cos x\right)\left(\sin x+\cos x\right)\right|\)
\(\sin x+\cos x=m\Rightarrow\cos x=m-\sin x\Rightarrow\sin x-\cos x=\sin x-m+\sin x=2\sin x-m\)
Có \(\sin x+\cos x=m\Rightarrow\sin^2x+\cos^2x+2\sin x.\cos x=m^2\)
\(\Leftrightarrow2\sin x.\cos x=m^2-1\)
\(\left(\sin x-\cos x\right)^2=\sin^2x+\cos^2x-2\sin x.\cos x=1-2.\left(m^2-1\right)=1-2m^2+2=3-2m^2\)
\(\Rightarrow\sin x-\cos x=\sqrt{\left(\sin x-\cos x\right)^2}=\sqrt{3-2m^2}\)
\(A=\left|m\sqrt{3-2m^2}\right|=\left|m\right|.\left|\sqrt{3-2m^2}\right|\)
P/s: lm đc mỗi đến đây thui à, cái CM kia chịu nhoa :)
\(\left(sinx+cosx\right)^2=m^2\Rightarrow1+2sinx.cosx=m^2\)\(\Rightarrow2sinx.cosx=m^2-1\)
\(\Rightarrow\left(sinx-cosx\right)^2=\left(sinx+cosx\right)^2-4sinx.cosx=m^2-2\left(m^2-1\right)=2-m^2\)
Mà \(\left(sinx-cosx\right)^2\ge0\) \(\forall x\Rightarrow2-m^2\ge0\Rightarrow m^2\le2\Rightarrow\left|m\right|\le\sqrt{2}\)
Ta lại có \(\left(sinx-cosx\right)^2=2-m^2\Rightarrow\left|sinx-cosx\right|=\sqrt{2-m^2}\)
\(A=\left|sin^4x-cos^4x\right|=\left|\left(sin^2x+cos^2x\right)\left(sin^2x-cos^2x\right)\right|\)
\(=\left|\left(sinx-cosx\right)\left(sinx+cosx\right)\right|\)
\(=\left|m\sqrt{2-m^2}\right|=\left|m\right|\sqrt{2-m^2}\)
a) \(\left(sinx+cosx\right)^2=sin^2x+2sinxcosx+cos^2x\)\(=1+2sinxcosx\).
b) \(\left(sinx-cosx\right)^2=sin^2x-2sinxcosx+cos^2x\)\(=1-2sinxcosx\).
c) \(sin^4x+cos^4x=\left(sin^2x+cos^2x\right)^2-2sin^2xcos^2x\)
\(=1-2sin^2xcos^2x\).
a.
\(\sqrt{2}sin\left(2x+\dfrac{\pi}{4}\right)=3sinx+cosx+2\)
\(\Leftrightarrow sin2x+cos2x=3sinx+cosx+2\)
\(\Leftrightarrow2sinx.cosx-3sinx+2cos^2x-cosx-3=0\)
\(\Leftrightarrow sinx\left(2cosx-3\right)+\left(cosx+1\right)\left(2cosx-3\right)=0\)
\(\Leftrightarrow\left(2cosx-3\right)\left(sinx+cosx+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=\dfrac{3}{2}\left(vn\right)\\sinx+cosx+1=0\end{matrix}\right.\)
\(\Rightarrow\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)=-1\)
\(\Leftrightarrow sin\left(x+\dfrac{\pi}{4}\right)=-\dfrac{\sqrt{2}}{2}\)
\(\Leftrightarrow...\)
b.
ĐKXĐ: \(cosx\ne\dfrac{1}{2}\Rightarrow\left[{}\begin{matrix}x\ne\dfrac{\pi}{3}+k2\pi\\x\ne-\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\)
\(\dfrac{\left(2-\sqrt{3}\right)cosx-2sin^2\left(\dfrac{x}{2}-\dfrac{\pi}{4}\right)}{2cosx-1}=1\)
\(\Rightarrow\left(2-\sqrt{3}\right)cosx+cos\left(x-\dfrac{\pi}{2}\right)=2cosx\)
\(\Leftrightarrow-\sqrt{3}cosx+sinx=0\)
\(\Leftrightarrow sin\left(x-\dfrac{\pi}{3}\right)=0\)
\(\Rightarrow x-\dfrac{\pi}{3}=k\pi\)
\(\Rightarrow x=\dfrac{\pi}{3}+k\pi\)
Kết hợp ĐKXĐ \(\Rightarrow x=\dfrac{4\pi}{3}+k2\pi\)
ta có : \(\dfrac{sinx+sin\left(\dfrac{x}{2}\right)}{1+cosx+cos\left(\dfrac{x}{2}\right)}=\dfrac{2sin\left(\dfrac{x}{2}\right).cos\left(\dfrac{x}{2}\right)+sin\left(\dfrac{x}{2}\right)}{2cos^2\left(\dfrac{x}{2}\right)+cos\left(\dfrac{x}{2}\right)}\)
\(=\dfrac{sin\left(\dfrac{x}{2}\right)\left(2cos\left(\dfrac{x}{2}\right)+1\right)}{cos\left(\dfrac{x}{2}\right)\left(2cos\left(\dfrac{x}{2}\right)+1\right)}=\dfrac{sin\left(\dfrac{x}{2}\right)}{cos\left(\dfrac{x}{2}\right)}=tan\left(\dfrac{x}{2}\right)\left(đpcm\right)\)
Ta có :
\(\frac{sinx+sin\left(\frac{x}{2}\right)}{1+cosx+cos\left(\frac{x}{2}\right)}=\frac{2sin\left(\frac{x}{2}\right).cos\left(\frac{x}{2}\right)+sin\left(\frac{x}{2}\right)}{2cos^2\left(\frac{x}{2}\right)+cos\left(\frac{x}{2}\right)}\)
\(=\frac{sin\left(\frac{x}{2}\right)\left(2cos\left(\frac{x}{2}\right)+1\right)}{cos\left(\frac{x}{2}\right)\left(2cos\left(\frac{x}{2}\right)+1\right)}=\frac{sin\left(\frac{x}{2}\right)}{cos\left(\frac{x}{2}\right)}\)
\(=tan\left(\frac{x}{2}\right)\left(đpcm\right)\)
1,\(A=3\left(sin^4x+cos^4x\right)-2\left(sin^2x+cos^2x\right)\left(sin^4x-sin^2x.cos^2x+cos^4x\right)\)
\(=3\left(sin^4x+cos^4x\right)-2\left(sin^4x-sin^2x.cos^4x+cos^4x\right)\)
\(=sin^4x+2sin^2x.cos^2x+cos^4x=\left(sin^2x+cos^2x\right)^2=1\)
Vậy...
2,\(B=cos^6x+2sin^4x\left(1-sin^2x\right)+3\left(1-cos^2x\right)cos^4x+sin^4x\)
\(=-2cos^6x+3sin^4x-2sin^6x+3cos^4x\)
\(=-2\left(sin^2x+cos^2x\right)\left(sin^4x-sin^2x.cos^2x+cos^4x\right)+3\left(cos^4x+sin^4x\right)\)
\(=-2\left(sin^4x-sin^2x.cos^2x+cos^4x\right)+3\left(cos^4x+sin^4x\right)\)\(=cos^4x+sin^4x+2sin^2x.cos^2x=1\)
Vậy...
3,\(C=\dfrac{1}{2}\left[cos\left(-\dfrac{7\pi}{12}\right)+cos\left(2x-\dfrac{\pi}{12}\right)\right]+\dfrac{1}{2}\left[cos\left(-\dfrac{7\pi}{12}\right)+cos\left(2x+\dfrac{11\pi}{12}\right)\right]\)
\(=cos\left(-\dfrac{7\pi}{12}\right)+\dfrac{1}{2}\left[cos\left(2x-\dfrac{\pi}{12}\right)+cos\left(2x+\dfrac{11\pi}{12}\right)\right]\)\(=\dfrac{-\sqrt{6}+\sqrt{2}}{4}+\dfrac{1}{2}\left[cos\left(2x-\dfrac{\pi}{12}\right)+cos\left(2x-\dfrac{\pi}{12}+\pi\right)\right]\)
\(=\dfrac{-\sqrt{6}+\sqrt{2}}{4}+\dfrac{1}{2}\left[cos\left(2x-\dfrac{\pi}{12}\right)-cos\left(2x-\dfrac{\pi}{12}\right)\right]\)\(=\dfrac{-\sqrt{6}+\sqrt{2}}{4}\)
Vậy...
4, \(D=cos^2x+\left(-\dfrac{1}{2}cosx-\dfrac{\sqrt{3}}{2}sinx\right)^2+\left(-\dfrac{1}{2}.cosx+\dfrac{\sqrt{3}}{2}.sinx\right)^2\)
\(=cos^2x+\dfrac{1}{4}cos^2x+\dfrac{\sqrt{3}}{4}cosx.sinx+\dfrac{3}{4}sin^2x+\dfrac{1}{4}cos^2x-\dfrac{\sqrt{3}}{4}cosx.sinx+\dfrac{3}{4}sin^2x\)
\(=\dfrac{3}{2}\left(cos^2x+sin^2x\right)=\dfrac{3}{2}\)
Vậy...
5, Xem lại đề
6,\(F=-cosx+cosx-tan\left(\dfrac{\pi}{2}+x\right).cot\left(\pi+\dfrac{\pi}{2}-x\right)\)
\(=tan\left(\pi-\dfrac{\pi}{2}-x\right).cot\left(\dfrac{\pi}{2}-x\right)\)\(=tan\left(\dfrac{\pi}{2}-x\right).cot\left(\dfrac{\pi}{2}-x\right)\)\(=cotx.tanx=1\)
Vậy...
a) Ta có (\sin x+\cos x)^{2}=\sin ^{2} x+2 \sin x \cos x+\cos ^{2} x=1+2 \sin x \cos x(sinx+cosx)2=sin2x+2sinxcosx+cos2x=1+2sinxcosx (*)
Mặt khác \sin x+\cos x=msinx+cosx=m nên m^{2}=1+2 \sin \alpha \cos \alpham2=1+2sinαcosα hay \sin \alpha \cos \alpha=\dfrac{m^{2}-1}{2}sinαcosα=2m2−1
Đặt A=\left|\sin ^{4} x-\cos ^{4} x\right|A=∣∣sin4x−cos4x∣∣. Ta có
A=\left|\left(\sin ^{2} x+\cos ^{2} x\right)\left(\sin ^{2} x-\cos ^{2} x\right)\right|=|(\sin x+\cos x)(\sin x-\cos x)|A=∣∣(sin2x+cos2x)(sin2x−cos2x)∣∣=∣(sinx+cosx)(sinx−cosx)∣
\Rightarrow A^{2}=(\sin x+\cos x)^{2}(\sin x-\cos x)^{2}=(1+2 \sin x \cos x)(1-2 \sin x \cos x)⇒A2=(sinx+cosx)2(sinx−cosx)2=(1+2sinxcosx)(1−2sinxcosx)
\Rightarrow A^{2}=\left(1+\dfrac{m^{2}-1}{2}\right)\left(1-\dfrac{m^{2}-1}{2}\right)=\dfrac{3+2 m^{2}-m^{4}}{4}⇒A2=(1+2m2−1)(1−2m2−1)=43+2m2−m4
Vậy A=\dfrac{\sqrt{3+2 m^{2}-m^{4}}}{2}A=23+2m2−m4
b) Ta có 2 \sin x \cos x \leq \sin ^{2} x+\cos ^{2} x=12sinxcosx≤sin2x+cos2x=1 kết hợp với (*)(∗) suy ra
(\sin x+\cos x)^{2} \leq 2 \Rightarrow|\sin x+\cos x| \leq \sqrt{2}(sinx+cosx)2≤2⇒∣sinx+cosx∣≤2
Vậy |m| \leq \sqrt{2}∣m∣≤2.
Đặt \(\left|A\right|=\left|mcos\left(x\right)+sin\left(x\right)\right|\)
\(\Rightarrow A^2=\left[m.cos\left(x\right)+1.sin\left(x\right)\right]^2\le\left(m^2+1^2\right)\left[cos^2\left(x\right)+sin^2\left(x\right)\right]\)
\(\Rightarrow A^2\le m^2+1\) (Vì cos2x + sin2x = 1)
\(\Rightarrow\left|A\right|\le\sqrt{m^2+1}\) hay \(\left|mcos\left(x\right)+sin\left(x\right)\right|\le\sqrt{m^2+1}\) (đpcm)
thanks bạn nha, trả lời nhanh ghê!!