Question

If $\theta$ is an acute angle and $\tan \theta +\cot \theta =2$, find the value of : ${\tan }^7\theta +{\cot }^7\theta$.

Answer 1

$\tan \theta +\cot \theta =2$

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

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

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

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

$\tan \theta =1$

$\theta ={45}^o$

Therefore, ${\tan }^7\theta +{\cot }^7\theta =1+1=2$