Question

If $\sin \theta +\text{cosec}\theta =2$, then the value of ${\sin }^6\theta +{\text{cosec}}^6\theta$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. $2^3$
• E. $2^6$

Answer 1

The correct option is C $2$

Given that, $\sin \theta +\text{cosec}\theta =2$

Squaring both sides, we get

$(\sin \theta +\text{cosec}\theta {)}^2=2^2$

$\Rightarrow {\sin }^2\theta +{\text{cosec}}^2\theta +2=4$

$\Rightarrow {\sin }^2\theta +{\text{cosec}}^2\theta =2$

Cubing both sides, we get

$\Rightarrow ({\sin }^2\theta +{\text{cosec}}^2\theta {)}^3=2^3$

$\Rightarrow {\sin }^6\theta +{\text{cosec}}^6\theta +3{\sin }^2\theta .{\text{cosec}}^2\theta ({\sin }^2\theta +{\text{cosec}}^2\theta )=8$

$\Rightarrow {\sin }^6\theta +{\text{cosec}}^6\theta +6=8$

$\Rightarrow {\sin }^6\theta +{\text{cosec}}^6\theta =2$

Hence, C is the correct option.