If tanθ + cotθ = -2, then the value of tan⁹θ + cot⁹θ is:

Solution (By Examveda Team)

$$\eqalign{ & \tan \theta + \cot \theta = - 2 \cr & \Rightarrow \tan \theta + \frac{1}{{\tan \theta }} = - 2 \cr & \Rightarrow \tan \theta = - 1 \cr & \Rightarrow {\tan ^9}\theta + {\cot ^9}\theta \cr & = {\tan ^9}\theta + \frac{1}{{{{\tan }^9}\theta }} \cr & = {\left( { - 1} \right)^9} + \frac{1}{{{{\left( { - 1} \right)}^9}}} \cr & = - 1 - 1 \cr & = - 2 \cr} $$