\(tanα+cotα=2\) হলে \(tan^{13}α+cot^{13}α\) -এর মান— (a) 13 (b) 2 (c) 1 (d) 0 Madhyamik 2019
Answer: B
\(tan\alpha+cot\alpha=2\)
বা, \(tan\alpha+\cfrac{1}{tan\alpha}=2\)
বা, \(\cfrac{tan^2\alpha+1}{tan\alpha}=2\)
বা, \(tan^2\alpha+1=2tan\alpha\)
বা, \(tan^2\alpha+1-2tan\alpha=0\)
বা, \((tan\alpha-1)^2=0\)
বা, \(tan\alpha-1=0\)
বা, \(tan\alpha=1\)
\(\therefore cot\alpha=1\)
\(\therefore tan^{13}α+cot^{13}α =1^{13}+1^{13}=1+1=2\)
\(tan\alpha+cot\alpha=2\)
বা, \(tan\alpha+\cfrac{1}{tan\alpha}=2\)
বা, \(\cfrac{tan^2\alpha+1}{tan\alpha}=2\)
বা, \(tan^2\alpha+1=2tan\alpha\)
বা, \(tan^2\alpha+1-2tan\alpha=0\)
বা, \((tan\alpha-1)^2=0\)
বা, \(tan\alpha-1=0\)
বা, \(tan\alpha=1\)
\(\therefore cot\alpha=1\)
\(\therefore tan^{13}α+cot^{13}α =1^{13}+1^{13}=1+1=2\)