\(sina=\frac{4}{5}\Rightarrow sin^2a=\frac{16}{25}\Rightarrow cos^2a=1-sin^2a=\frac{9}{25}\)
\(A=tan^2a-2cot^2a=\frac{sin^2a}{cos^2a}-\frac{2cos^2a}{sin^2a}\)
\(A=\frac{\frac{16}{25}}{\frac{9}{25}}-\frac{2.\frac{9}{25}}{\frac{16}{25}}=\frac{16}{9}-\frac{9}{8}=\frac{47}{72}\)
Ta có: \(\sin^2\alpha+\cos^2\alpha=1\)
\(\Rightarrow\cos^2\alpha=1-\sin^2\alpha=1-\left(\frac{4}{5}\right)^2=\frac{9}{25}\)
\(\Rightarrow\cos\alpha=\frac{3}{5}\)
Lại có: \(tan\alpha=\frac{\sin\alpha}{\cos\alpha}=\frac{\frac{4}{5}}{\frac{3}{5}}=\frac{4}{3}\)
\(\Rightarrow\tan^2\alpha=\frac{16}{9}\)
Mà \(\tan\alpha.\cot\alpha=1\Rightarrow\cot\alpha=\frac{1}{\tan\alpha}=\frac{1}{\frac{4}{3}}=\frac{3}{4}\Rightarrow\cot^2\alpha=\frac{9}{16}\)
\(P=\tan^2\alpha-2\cot^2\alpha\)
\(=\frac{16}{9}-2.\left(\frac{9}{16}\right)\)
\(=\frac{16}{9}-\frac{9}{8}\)
\(=\frac{47}{72}\)
