Question

Cho \(\tan a-\cot a=3\) . Tính \(\tan^2a+\cot^2a\) và \(\tan^4a-\cot^4a\)

Answer 1

\(tana-cota=3\Rightarrow\left(tana-cota\right)^2=9\)

\(\Rightarrow tan^2a+cot^2a-2=9\Rightarrow tan^2a+cot^2a=11\)

\(tan^2a+cot^2a+2=13\Rightarrow\left(tana+cota\right)^2=13\)

\(\Rightarrow tana+cota=\pm\sqrt{13}\)

- Với \(tana+cota=\sqrt{13}\)

\(\Rightarrow tan^4a-cot^4a=\left(tana-cota\right)\left(tana+cota\right)\left(tan^2a+cot^2a\right)=3.\sqrt{13}.11=...\)

- Với \(tana+cota=-\sqrt{13}\Rightarrow tan^4a-cot^4a=-\sqrt{13}.3.11=...\)