Question

If $\tan \theta =4,$ then $\left(\frac{\tan \theta }{\frac{{\sin }^3\theta }{\cos \theta }+\sin \theta \cos \theta }\right)$ is equal to

• A. $0$
• B. $2\sqrt{2}$
• C. $\sqrt{2}$
• D. $1$

Answer 1

The correct option is D $1$

Given, $\tan \theta =4,$
$\Rightarrow \frac{\tan \theta }{\frac{{\sin }^3\theta }{\cos \theta }+\sin \theta \cos \theta }=\frac{\tan \theta }{\tan \theta {\sin }^2\theta +\frac{\sin \theta {\cos }^2\theta }{\cos \theta }}$
$=\frac{\tan \theta }{\tan \theta {\sin }^2\theta +\tan \theta {\cos }^2\theta }$
$=\frac{4}{4{\sin }^2\theta +4{\cos }^2\theta }$
$=\frac{4}{4({\sin }^2\theta +{\cos }^2\theta )}$
$=\frac{4}{4\left(1\right)}=1$ ($\because {\sin }^2\theta +{\cos }^2\theta =1$ )
Option D is correct.