\(a,\)
\(A=\left(sin\alpha+cos\alpha\right)^2+\left(sin\alpha-cos\alpha\right)^2\)
\(=sin^2\alpha+cos^2\alpha+2.sin\alpha.cos\alpha+sin^2\alpha+cos^2\alpha-2.sin\alpha.cos\alpha\)
\(=1+1=2\)
\(b,\)
\(VT=\dfrac{sin^2\alpha+cos^2\alpha-2.sin\alpha.cos\alpha-1}{tan\alpha-sin\alpha.cos\alpha}\)
\(=\dfrac{-2.sin\alpha.cos\alpha}{tan\alpha.sin^2\alpha}\)
\(=\dfrac{-2.cos\alpha}{tan\alpha.sin\alpha}\)
\(=-2.\dfrac{cos\alpha}{sin\alpha}.\dfrac{1}{tan\alpha}\)
\(=\dfrac{-2.cot\alpha}{tan\alpha}\)
\(=-2.cot^2\alpha=VP\) (điều phải chứng minh)
\(c,\)
\(VT^2=\dfrac{1+sin\alpha}{1-sin\alpha}+\dfrac{1-sin\alpha}{1+sin\alpha}+2\)
\(=\dfrac{\left(1+sin\alpha\right)^2+\left(1-sin\alpha\right)^2+2.\left(1+sin\alpha\right).\left(1-sin\alpha\right)}{\left(1-sin\alpha\right)\left(1+sin\alpha\right)}\)
\(=\dfrac{\left(1+sin\alpha+1-sin\alpha\right)^2}{1-sin^2\alpha}\)
\(=\dfrac{4}{cos^2\alpha}\)
\(\Rightarrow VT=\pm\dfrac{2}{cos\alpha}\)
Vì \(0< \alpha< \dfrac{\pi}{2}\Rightarrow cos\alpha>0\Rightarrow VT=\dfrac{2}{cos\alpha}=VP\) (điều phải chứng minh)
