Question

From the options given below which option is equal to $\sin 4\theta$.

• A. $8{\sin }^4\theta -8{\sin }^2\theta +1$
• B. $8{\cos }^4\theta -8{\cot }^2\theta +1$
• C. $4{\cos }^3\theta \sin \theta -4\cos \theta {\sin }^3\theta$
• D. $4{\sin }^3\theta \cos \theta -4\sin \theta {\cos }^3\theta$

Answer 1

Answer: (C)

$4{\cos }^3\theta \sin \theta -4\cos \theta {\sin }^3\theta$

$\sin 4\theta =2\sin 2\theta \cos 2\theta$

$=2\times 2\sin \theta \cos \theta ({\cos }^2\theta -{\sin }^2\theta )$

$=4{\cos }^3\theta \sin \theta -4\cos \theta {\sin }^3\theta$