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\(\frac{1+sin^2x}{1-sin^2x}=\frac{cos^2x+sin^2x+sin^2x}{cos^2x+sin^2x-sin^2x}=\frac{cos^2x+2sin^2x}{cos^2x}=1+2\left(\frac{sinx}{cosx}\right)^2=1+2tan^2x\)
Xét tam giác ABC vuông tại A có AH là đường cao và AM là trung tuyến
Đặt \(\widehat{MAC}=\widehat{MCA}=x\)thì \(\widehat{BMA}=2x\)(theo tính chất đường trung tuyến ứng với cạnh huyền của tam giác vuông)
a) Ta có: \(\sin2x=\frac{AH}{AM}=2.\frac{AH}{BC}=2.\frac{AH}{AC}.\frac{AC}{BC}=2.\sin ACH.\cos ACB=2\cos x.\sin x\)
b) \(\cos2x=\frac{HM}{AM}=\frac{2HM}{BC}=\frac{2HC-2CM}{BC}=2.\frac{HC}{BC}-1=2.\frac{HC}{ AC}.\frac{AC}{BC}-1=2.\cos ACH.\cos ACB-1=2\cos^2x-1=2\cos^2x-\left(\sin^2x+\cos^2x\right)=\cos^2x-\sin^2x\)c) \(\tan2x=\frac{\sin2x}{\cos2x}=\frac{2\cos x.\sin x}{\cos^2x-\sin^2x}=\frac{2.\frac{\sin x}{\cos x}}{\frac{\cos^2x}{\cos^2x}-\frac{\sin^2x}{\cos^2x}}=\frac{2\tan x}{1-\tan^2x}\)
Chứng minh đẳng thức:
\(\frac{\sin^2x}{\sin x-\cos x}-\frac{\sin x+\cos x}{\tan^2x-1}=\sin x+\cos x\)
\(\frac{\sin^2x}{\sin x-\cos x}-\frac{\sin x+\cos x}{\tan^2x-1}\)
\(=\frac{\sin^2x}{\sin x-\cos x}-\frac{\sin x+\cos x}{\frac{\sin^2x-\cos^2x}{\cos^2x}}\)
\(=\frac{\sin^2x}{\sin x-\cos x}-\frac{\cos^2x}{\sin x-\cos x}=\sin x+\cos x\)
Xong
a) \(\dfrac{1}{1+tan\alpha}+\dfrac{1}{1+cot\alpha}\)
\(=\dfrac{1}{1+\dfrac{1}{cot\alpha}}+\dfrac{1}{1+cot\alpha}\)
\(=\dfrac{1}{\dfrac{cot\alpha+1}{cot\alpha}}+\dfrac{1}{1+cot\alpha}\)
\(=\dfrac{cot\alpha}{cot\alpha+1}+\dfrac{1}{1+cot\alpha}\)
\(=\dfrac{cot\alpha+1}{cot\alpha+1}=1\) (đpcm)
b) \(tan^2x+cot^2x+2\)
\(=\dfrac{sin^2x}{cos^2x}+\dfrac{cos^2x}{sin^2x}+2\)
\(=\dfrac{sin^2x}{cos^2x}+1+\dfrac{cos^2x}{sin^2x}+1\)
\(=\dfrac{sin^2x+cos^2x}{cos^2x}+\dfrac{cos^2x+sin^2x}{sin^2x}\)
\(=\dfrac{1}{cos^2x}+\dfrac{1}{sin^2x}\) (đpcm)
c) \(sinx.cosx.\left(1+tanx\right)\left(1+cotx\right)\)
\(=\left(sinx.cosx+sinx.cosx.tanx\right)\left(1+cotx\right)\)
\(=\left(sinx.cosx+sinx.cosx.\dfrac{sinx}{cosx}\right)\left(1+cotx\right)\)
\(=\left(sinx.cosx+sin^2x\right)\left(1+cotx\right)\)
\(=\left(sinx.cosx+sin^2x\right)\left(1+\dfrac{cosx}{sinx}\right)\)
\(=sinx.cosx+cos^2x+sin^2x+sinx.cosx\)
\(=1+sin^2x.cos^2x\)
Câu cuối không biết chỗ sai, mong mọi người chỉ bảo ạ ^^
\(A=s\left(x\right)cs\left(x\right)+\frac{\left(s^3\left(x\right)+cs^3\left(x\right)\right)}{cs\left(x\right)\left(1+t\left(x\right)\right)}=s\left(x\right)cs\left(x\right)+\left(\frac{\left(s\left(x\right)+cs\left(x\right)\right)\left(1-s\left(x\right)cs\left(x\right)\right)}{\left(s\left(x\right)+cs\left(x\right)\right)}\right)\)
\(=1\) vì \(s\left(x\right)+cs\left(x\right)\ne0,\forall0< =x< =\frac{\pi}{2}\)
a) \(cos^4x-sin^4x=\left(cos^2x+sin^2x\right)\left(cos^2x-sin^2x\right)=cos^2x-sin^2x\)
b) \(\frac{1}{1+tanx}+\frac{1}{1+cotx}=\frac{1}{1+tanx}+\frac{tanxcotx}{tanxcotx+cotx}=\frac{1}{1+tanx}+\frac{tanx}{tanx+1}\)
\(=\frac{1+tanx}{1+tanx}=1\)
c) Ta có: \(1+tan^2x=1+\frac{sin^2x}{cos^2x}=\frac{cos^2x+sin^2x}{cos^2x}=\frac{1}{cos^2x}\)
\(\Rightarrow\frac{1}{1+tan^2x}=cos^2x\)
Tương tự \(\frac{1}{1+tan^2y}=cos^2y\)
\(\Rightarrow cos^2x-cos^2y=\frac{1}{1+tan^2x}-\frac{1}{1+tan^2y}\)
\(cos^2x-cos^2y=\left(1-sin^2x\right)-\left(1-sin^2y\right)=sin^2y-sin^2x\)
d) \(\frac{1+sin^2x}{1-sin^2x}=\frac{cos^2x+sin^2x+sin^2x}{cos^2x+sin^2x-sin^2x}=\frac{cos^2x+2sin^2x}{cos^2x}=1+2\left(\frac{sinx}{cosx}\right)^2=1+2tan^2x\)