Question : Simplify the following expression.
$\frac{\sin \theta - 2 \sin ^3 \theta}{2 \cos ^3 \theta - \cos \theta}$
Option 1: $\tan \theta$
Option 2: $\sin \theta$
Option 3: $\sec \theta$
Option 4: $\cos \theta$
Correct Answer: $\tan \theta$
Solution :
Given: $\frac{\sin \theta - 2 \sin ^3 \theta}{2 \cos ^3 \theta - \cos \theta}$
$=\frac{\sin \theta(1 -2 \sin ^2\theta)}{\cos \theta( 2 \cos ^2\theta - 1)}$
$=\frac{\sin \theta(\sin ^2\theta + \cos^2\theta - 2 \sin ^2\theta)}{\cos \theta( 2\cos ^2\theta - \sin ^2\theta - \cos^2\theta)}$ [as $\sin ^2\theta +\cos^2\theta = 1$]
$=\frac{\sin\theta(\cos^{2}\theta - \sin^{2}\theta)}{\cos\theta(\cos^{2}\theta - \sin^{2}\theta)}$
$=\tan \theta$
Hence, the correct answer is $\tan \theta$.
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