Question

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

Answer 1

$tan\theta +cot\theta =2$

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

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

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

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

$\Rightarrow tan\theta -1=0\Rightarrow tan\theta =1\Rightarrow tan\theta =tan{45}^{\circ }\Rightarrow \theta =45$

$\therefore ta{n}^7\theta +co{t}^7\theta =ta{n}^7{45}^{\circ }+co{t}^7{45}^{\circ }=(tan{45}^{\circ }{)}^7+(cot{45}^{\circ }{)}^7=(1{)}^7+(1{)}^7=2$