Question

If $\theta$ is an acute angle and $\tan \theta +\cot \theta =2$, find the value of ${\tan }^9\theta +{\cot }^9\theta$

• A. $1$
• B. $\sqrt{2}$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

We have $\tan \theta +\cot \theta =2$
$\Rightarrow \tan \theta +\frac{1}{\tan \theta }=2$

$\Rightarrow \frac{{\tan }^2\theta +1}{\tan \theta }=2$
$\Rightarrow {\tan }^2\theta +1=2\tan \theta$

$\Rightarrow {\tan }^2\theta -2\tan \theta +1=0$
$\Rightarrow {(\tan \theta -1)}^2=0$

$\Rightarrow \tan \theta -1=0$

$\Rightarrow \tan \theta =1$

$\Rightarrow \tan \theta =\tan {45}^o$

$\Rightarrow \theta ={45}^o$
$\therefore {\tan }^9\theta +{\cot }^9{45}^o={\tan }^9{45}^o+{\cot }^9{45}^o={\left(\tan {45}^o\right)}^9+{\left(\cot {45}^o\right)}^9$
$=1^9+1^9=2$