Question

$\sin \theta +\csc \theta =2$ then find the value of ${\sin }^5\theta +{\csc }^6\theta$

Answer 1

$\sin \theta +\csc \theta =2\left(\text{Given}\right)$

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

$\Rightarrow {\sin }^2\theta +1=2\sin \theta$

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

$\Rightarrow \sin \theta =1$

Therefore,

${\sin }^5\theta +{\csc }^6\theta$

$={\left(1\right)}^5+{\left(1\right)}^6$

$=1+1=2$