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\(A=\frac{1-2sina.cosa}{sin^2a-cos^2a}=\frac{sin^2a+cos^2a-2sina.cosa}{\left(sina-cosa\right)\left(sina+cosa\right)}=\frac{\left(sina-cosa\right)^2}{\left(sina-cosa\right)\left(sina+cosa\right)}=\frac{sina-cosa}{sina+cosa}\)
b/ \(A=\frac{\frac{sina}{cosa}-\frac{cosa}{cosa}}{\frac{sina}{cosa}+\frac{cosa}{cosa}}=\frac{tana-1}{tana+1}=\frac{\frac{1}{3}-1}{\frac{1}{3}+1}=-\frac{1}{2}\)
bài 1
a) \(M=\sin^242^o+\sin^243^o+\sin^244^o+\sin^245^o+\sin^246^o+\sin^247^o+\sin^248^o\)
\(M=\cos^248^o+\cos^247^o+\cos^246^o+\sin^245^o+\sin^246^o+\sin^247^o+\sin^248^o\)
\(M=\left(\sin^248^o+\cos^248^o\right)+\left(\sin^247^o+\cos^247^o\right)+\left(\sin^246^o+\cos^246^o\right)+\sin^245^o\)
\(M=1+1+1+0,5\)
\(M=3,5\)
bài 1
b) \(N=\cos^215^o-\cos^225^o+\cos^235^o-\cos^245^o+\cos^255^o-\cos^265^o+\cos^275^o\)
\(N=\sin^275^o-\sin^265^o+\sin^255^o-\cos^245^o+\cos^255^o-\cos^265^o+\cos^275^o\)
\(N=\left(\sin^275^o+\cos^275^o\right)-\left(\sin^265^o+\cos^265^o\right)+\left(\sin^255^o+\cos^255^o\right)-\cos^245^o\)
\(N=1-1+1-0,5\)
\(N=0,5\)
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) \(\frac{1+sin^2x}{1-sin^2x}=\frac{cos^2x+sin^2x+sin^2x}{cos^2x+sin^2x-sin^2x}=\frac{cos^2x+2sin^2x}{cos^2x}=1+2\left(\frac{sinx}{cosx}\right)^2=1+2tan^2x\)
1: \(sin^6x+cos^6x+3sin^2x\cdot cos^2x\)
\(=\left(sin^2x+cos^2x\right)^2-3\cdot sin^2x\cdot cos^2x\cdot\left(sin^2x+cos^2x\right)+3\cdot sin^2x\cdot cos^2x\)
=1
2: \(sin^4x-cos^4x\)
\(=\left(sin^2x+cos^2x\right)\left(sin^2x-cos^2x\right)\)
\(=1-2\cdot cos^2x\)