Question

If $\tan \alpha =2$, then the value of $\frac{\sin \alpha }{{\sin }^3\alpha +{\cos }^3\alpha }$ is

• A. $\frac{2}{9}$
• B. $\frac{5}{9}$
• C. $\frac{10}{9}$
• D. $\frac{5.5}{9}$

Answer 1

Answer: (C)

$\frac{10}{9}$

Given equation,

$\frac{\sin \alpha }{(\sin \alpha {)}^3+(\cos \alpha {)}^3}$

$=>$ divide numerator and denominator with $(\cos \alpha {)}^3$

$=>\frac{\frac{\sin \alpha }{\cos \alpha }\times \frac{1}{(\cos \alpha {)}^2}}{(tan\alpha {)}^3+1}$

$=>\frac{\tan \alpha \times (sec\alpha {)}^2}{(\tan \alpha {)}^3+1}$

$=>\frac{\tan \alpha \times \left(\right(tan\alpha {)}^2+1)}{(\tan \alpha {)}^3+1}$

put the value of $\tan \alpha =2,$

$=>\frac{2\times (2^2+1)}{2^3+1}$

$=>\frac{10}{9}$