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\(B=\left(sina+cosa\right)^2-\left(cosa-sina\right)^2=\left(sin^2a+2sinacosa+cos^2a\right)-\left(cos^2a-2cosasina+sin^2a\right)=sin^2a+2sinacosa+cos^2a-cos^2a+2cosasina-sin^2a=4sinacosa\)\(A=\dfrac{1+2sinacosa}{sina+cosa}=\dfrac{sin^2a+cos^2a+2cosasina}{sina+cosa}=\dfrac{\left(sina+cosa\right)^2}{sina+cosa}=sina+cosa\)
C mik bó tay
a, Sử dụng tích chéo:
Ta có:
+/ \(\cos\alpha.\cos\alpha=\cos^2\alpha\) (1)
+/ \(\left(1+\sin\alpha\right)\left(1-\sin\alpha\right)=1-\sin^2\alpha\)
Mà \(\sin^2\alpha+\cos^2\alpha=1\)
\(\Rightarrow1-\sin^2\alpha=\cos^2\alpha\)
hay \(\left(1+\sin\alpha\right)\left(1-\sin\alpha\right)=\cos^2\alpha\) (2)
Từ (1), (2)
\(\Rightarrow\)\(\cos\alpha.\cos\alpha=\)\(\left(1+\sin\alpha\right)\left(1-\sin\alpha\right)\)
\(\Rightarrow\)\(\dfrac{\cos\alpha}{1-\sin\alpha}=\dfrac{1+\sin\alpha}{\cos\alpha}\) (đpcm)
b/ xem lại đề
Lời giải:
\(M=\frac{(\cos a-\sin a)^2-(\cos a+\sin a)^2}{\cos a\sin a}\)
\(=\frac{\cos ^2a-2\sin a\cos a+\sin ^2a-(\cos ^2a+2\sin a\cos a+\sin ^2a)}{\cos a\sin a}\)
\(=\frac{-4\sin a\cos a}{\cos a\sin a}=-4\)
a) \(1+tan^2\alpha=1+\left(\dfrac{sin\alpha}{cos\alpha}\right)^2=\dfrac{sin^2\alpha+cos^2\alpha}{cos^2\alpha}=\dfrac{1}{cos^2\alpha}\)
b) \(1+cot^2\alpha=1+\left(\dfrac{cos\alpha}{sin\alpha}\right)^2=\dfrac{cos^2\alpha+sin^2\alpha}{sin^2\alpha}=\dfrac{1}{sin^2\alpha}\)
c) \(tan^2\alpha\left(2sin^2\alpha+3cos^2\alpha-2\right)=tan^2\alpha\left[cos^2\alpha+2\left(sin^2\alpha+cos^2\alpha\right)-2\right]=\dfrac{sin^2\alpha}{cos^2\alpha}\times cos^2\alpha=sin^2\alpha\)
a)
\(1+tan^2\alpha=1+\left(\dfrac{sin\alpha}{cos\alpha}\right)^2=\dfrac{cos^2\alpha+sin^2\alpha}{cos^2\alpha}=\dfrac{1}{cos^2\alpha}\)
b)\(1+cot^2\alpha=1+\left(\dfrac{cos\alpha}{sin\alpha}\right)^2=\dfrac{sin^2\alpha+cos^2\alpha}{sin^2\alpha}=\dfrac{1}{sin^2\alpha}\)
c) mình chưa rõ đề nha