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\(C=2sin3x.cos3x.\frac{cos3x}{sin3x}-\left(cos^23x-sin^23x\right)\)
\(=2cos^23x-cos^23x+sin^23x=cos^23x+sin^23x=1\)
\(\sqrt{2}sin\left(x-\frac{\pi}{4}\right)=\sqrt{2}\left(sinx.cos\frac{\pi}{4}-cosx.sin\frac{\pi}{4}\right)\)
\(=\sqrt{2}\left(sinx.sin\frac{\pi}{4}-cosx.cos\frac{\pi}{4}\right)=-\sqrt{2}\left(cosx.cos\frac{\pi}{4}-sinx.sin\frac{\pi}{4}\right)=-\sqrt{2}cos\left(x+\frac{\pi}{4}\right)\)
Câu này bạn ghi nhầm đề (lưu ý rằng \(sin\frac{\pi}{4}=cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\))
Câu 2b bạn cũng xem lại đề, chắc chắn ko đúng
\(\frac{A}{2}+\frac{B}{2}+\frac{C}{2}=90^0\Rightarrow sin\frac{A}{2}=cos\left(\frac{B}{2}+\frac{C}{2}\right)=cos\frac{B}{2}cos\frac{C}{2}-sin\frac{B}{2}sin\frac{C}{2}\)
Câu 3 bạn cũng ghi sai đề luôn
Trong 1 ngày đẹp trời thì câu 4 cũng sai luôn cho đỡ lạc lõng đồng đội:
\(sin\left(a+b-b\right)=sin\left(a+b\right)cosb-cos\left(a+b\right)sinb=2sin\left(a+b\right)\)
\(\Leftrightarrow sin\left(a+b\right)\left[cosb-2\right]=cos\left(a+b\right).sinb\)
\(\Leftrightarrow\frac{sin\left(a+b\right)}{cos\left(a+b\right)}=\frac{sinb}{cosb-2}\Leftrightarrow tan\left(a+b\right)=\frac{sinb}{cosb-2}\)
4 câu bạn ghi đúng đề bài duy nhất câu 1, kinh thiệt :(
\(A=\frac{2sinx.cosx+sinx}{1+2cos^2x-1+cosx}=\frac{sinx\left(2cosx+1\right)}{cosx\left(2cosx+1\right)}=\frac{sinx}{cosx}=tanx\)
\(B=\frac{cosa}{sina}\left(\frac{1+sin^2a}{cosa}-cosa\right)=\frac{cosa}{sina}\left(\frac{1+sin^2a-cos^2a}{cosa}\right)=\frac{cosa}{sina}.\frac{2sin^2a}{cosa}=2sina\)
\(C=\frac{1+cos2x+cosx+cos3x}{2cos^2x-1+cosx}=\frac{1+2cos^2x-1+2cos2x.cosx}{cos2x+cosx}=\frac{2cosx\left(cosx+cos2x\right)}{cos2x+cosx}=2cosx\)
\(D=\frac{2sinx.cosx.\left(-tanx\right)}{-tanx.sinx}-2cosx=2cosx-2cosx=0\)
\(E=cos^2x.cot^2x-cot^2x+cos^2x+2cos^2x+2sin^2x\)
\(E=cot^2x\left(cos^2x-1\right)+cos^2x+2=\frac{cos^2x}{sin^2x}\left(-sin^2x\right)+cos^2x+2=2\)
\(F=\frac{sin^2x\left(1+tan^2x\right)}{cos^2x\left(1+tan^2x\right)}=\frac{sin^2x}{cos^2x}=tan^2x\)
Câu G mẫu số có gì đó sai sai, sao lại là \(2sina-sina?\)
\(H=sin^4\left(\frac{\pi}{2}+a\right)-cos^4\left(\frac{3\pi}{2}-a\right)+1=cos^4a-sin^4a+1\)
\(=\left(cos^2a-sin^2a\right)\left(cos^2a+sin^2a\right)+1=cos^2a-\left(1-cos^2a\right)+1=2cos^2a\)
Thì tách bình thường thôi :)
\(A=\left[\tan\left(4\pi+\frac{\pi}{4}\right)+\tan\left(3\pi+\frac{\pi}{2}-x\right)\right]^2+\left[\cot\left(4\pi+\frac{\pi}{4}\right)+\cot\left(-x\right)\right]^2\)
\(A=\left[\tan\left(\frac{\pi}{4}\right)+\cot x\right]^2+\left[\cot\left(\frac{\pi}{4}\right)-\cot x\right]^2\)
\(A=\left(1+\cot x\right)^2+\left(1-\cot x\right)^2=...\)
a) \(sin6\alpha cot3\alpha cos6\alpha=2.sin3\alpha.cos3\alpha\dfrac{cos3\alpha}{sin3\alpha}-cos6\alpha\)
\(=2cos^23\alpha-\left(2cos^23\alpha-1\right)=1\) (Không phụ thuộc vào x).
b) \(\left[tan\left(90^o-\alpha\right)-cot\left(90^o+\alpha\right)\right]^2\)\(-\left[cot\left(180^o+\alpha\right)+cot\left(270^o+\alpha\right)\right]^2\)
\(=\left[cot\alpha+cot\left(90^o-\alpha\right)\right]^2\)\(-\left[cot\alpha+cot\left(90^o+\alpha\right)\right]^2\)
\(=\left[cot\alpha+tan\alpha\right]^2-\left[cot\alpha-tan\alpha\right]^2\)
\(=4tan\alpha cot\alpha=4\). (Không phụ thuộc vào \(\alpha\)).
\(a=\left(\frac{sina+\frac{sina}{cosa}}{cosa+1}\right)^2+1=\left(\frac{sina\left(cosa+1\right)}{cosa\left(cosa+1\right)}\right)^2+1\)
\(=tan^2a+1=\frac{1}{cos^2a}\)
\(b=\frac{sina}{cosa}\left(\frac{1+cos^2a-sin^2a}{sina}\right)=\frac{sina}{cosa}\left(\frac{2cos^2a}{sina}\right)=2cosa\)
\(c=1-\frac{cos^2a}{cot^2a}+\frac{sina.cosa}{\frac{cosa}{sina}}=1-cos^2a.\frac{sin^2a}{cos^2a}+\frac{sin^2a.cosa}{cosa}\)
\(=1-sin^2a+sin^2a=1\)
\(\frac{1-2sin^2x}{1-tanx}=\frac{cosx\left(1-2sin^2x\right)}{cosx-sinx}=\frac{cosx\left(cos^2x-sin^2x\right)}{cosx-sinx}=\frac{cosx\left(cosx+sinx\right)\left(cosx-sinx\right)}{cosx-sinx}\)
\(=cosx\left(cosx+sinx\right)=\frac{cosx\left(cosx+sinx\right)^2}{cosx+sinx}=\frac{cos^2x+sin^2x+2sinx.cosx}{1+\frac{sinx}{cosx}}=\frac{1+sin2x}{1+tanx}\)
\(\frac{x}{2}=a\Rightarrow\frac{cot^2a-cot^23a}{cos^2a.cos2a\left(1+cot^23a\right)}=\frac{sin^23a\left(cot^2a-cot^23a\right)}{cos^2a.cos2a}=\frac{sin^23a.cot^2a-cos^23a}{cos^2a.cos2a}\)
\(=\frac{sin^23a.cos^2a-cos^23a.sin^2a}{sin^2a.cos^2a.cos2a}=\frac{\left(sin3a.cosa-cos3a.sina\right)\left(sin3a.cosa+cos3a.sina\right)}{sin^2a.cos^2a.cos2a}\)
\(=\frac{sin\left(3a-a\right).sin\left(3a+a\right)}{sin^2a.cos^2a.cos2a}=\frac{sin2a.sin4a}{sin^2a.cos^2a.cos2a}=\frac{2sina.cosa.4sina.cosa.cos2a}{sin^2a.cos^2a.cos2a}\)
\(=\frac{8sin^2a.cos^2a.cos2a}{sin^2a.cos^2a.cos2a}=8\)
\(sin\left(a+b+a\right)=5sin\left(a+b-a\right)\)
\(\Leftrightarrow sin\left(a+b\right)cosa+cos\left(a+b\right).sina=5sin\left(a+b\right).cosa-5cos\left(a+b\right).sina\)
\(\Leftrightarrow6cos\left(a+b\right).sina=4sin\left(a+b\right).cosa\)
\(\Leftrightarrow\frac{2sin\left(a+b\right)cosa}{cos\left(a+b\right)sina}=3\Leftrightarrow\frac{2tan\left(a+b\right)}{tana}=3\)
\(cos5x.cos3x+sin7x.sinx=\frac{1}{2}cos8x+\frac{1}{2}cos2x-\frac{1}{2}cos8x+\frac{1}{2}cos6x\)
\(=\frac{1}{2}\left(cos6x+cos2x\right)=cos4x.cos2x\)
\(\frac{1-2sin^22x}{1-sin4x}=\frac{cos^22x-sin^22x}{cos^22x+sin^22x-2sin2x.cos2x}\)
\(=\frac{\left(cos2x-sin2x\right)\left(cos2x+sin2x\right)}{\left(cos2x-sin2x\right)^2}=\frac{cos2x+sin2x}{cos2x-sin2x}=\frac{\frac{cos2x}{cos2x}+\frac{sin2x}{cos2x}}{\frac{cos2x}{cos2x}-\frac{sin2x}{cos2x}}=\frac{1+tan2x}{1-tan2x}\)
\(2cosx-3cos\left(\pi-x\right)+5sin\left(4\pi-\frac{\pi}{2}-x\right)+cot\left(\pi+\frac{\pi}{2}-x\right)\)
\(=2cosx+3cosx-5sin\left(\frac{\pi}{2}+x\right)+cot\left(\frac{\pi}{2}-x\right)\)
\(=5cosx-5cosx+tanx=tanx\)