Question

If tan θ + cot θ = 2 and, 0° < θ < 90° then tan¹⁰ θ + cot¹⁰ θ is equal to _________.

Answer 1

$$\begin{gathered} \tan \theta +\cot \theta =2 \\ \Rightarrow \tan \theta +\frac{1}{\tan \theta }=2 \\ \Rightarrow \frac{{\tan }^2\theta +1}{\tan \theta }=2 \end{gathered}$$
$$\begin{gathered} \Rightarrow {\tan }^2\theta +1=2\tan \theta \\ \Rightarrow {\tan }^2\theta -2\tan \theta +1=0 \\ \Rightarrow {\left(\tan \theta -1\right)}^2=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow \tan \theta -1=0 \\ \Rightarrow \tan \theta =1 \\ \therefore \cot \theta =\frac{1}{\tan \theta }=1 \end{gathered}$$
So,
${\tan }^{10}\theta +{\cot }^{10}\theta ={\left(1\right)}^{10}+{\left(1\right)}^{10}=1+1=2$

If tan θ + cot θ = 2 and, 0° < θ < 90° then tan¹⁰ θ + cot¹⁰ θ is equal to ____2____.