Question

If tan $\alpha =2$ and $\alpha \in (\pi ,\frac{3\pi }{2})$ then the value of the expression $\frac{\cos \alpha }{{\sin }^3\alpha +{\cos }^3\alpha }$ is equal to ..................

• A. $\frac{5}{9}$
• B. 1
• C. $\frac{7}{9}$
• D. $\frac{5}{7}$

Answer 1

The correct option is A $\frac{5}{9}$

As $tan\alpha =2$ and $\alpha =\in (\pi ,\frac{3\pi }{2})\Rightarrow \alpha$ lies in 3rd quadrant in which both $\cos ,\sin$ are negative

Then $\cos \alpha =-\frac{1}{\sqrt{5}},\sin \alpha =-\frac{8}{5\sqrt{5}}$

Hence

$\frac{\cos a}{{\sin }^3\alpha +{\cos }^3\alpha }=\frac{-\frac{1}{\sqrt{5}}}{\frac{-8}{5\sqrt{5}}-\frac{1}{5\sqrt{5}}}=\frac{5}{9}$