Question

${\sin }^2\theta {\cos }^3\theta -{\sin }^4\theta \cos \theta$ is equal to

• A. $\frac{1}{2}{\sin }^{2}2\theta$
• B. $\frac{1}{4}\sin \theta \sin 4\theta$
• C. $\frac{1}{2}\cos \theta \sin 4\theta$
• D. $\frac{1}{4}\cos \theta \sin 4\theta$

Answer 1

The correct option is B $\frac{1}{4}\sin \theta \sin 4\theta$

${\sin }^2\theta {\cos }^3\theta -{\sin }^4\theta \cos \theta$
$={\sin }^2\theta \cos \theta ({\cos }^2\theta -{\sin }^2\theta )$
$={\sin }^2\theta \cos \theta \left(\cos 2\theta \right)$
$=\sin \theta \sin \theta \cos \theta \left(\cos 2\theta \right)$
$=\sin \theta \cdot \frac{2}{2}\sin \theta \cos \theta \left(\cos 2\theta \right)$
$=\frac{1}{2}\sin \theta \sin 2\theta \cos 2\theta$ $[\because 2\sin \theta \cos \theta =\sin 2\theta ]$
$=\frac{1}{2}\sin \theta \cdot \frac{2}{2}\sin 2\theta \cos 2\theta =\frac{1}{4}\sin \theta \cdot \sin 4\theta$