Question

Evaluate $\frac{{\sin }^3\theta +\sin 3\theta }{{\cos }^{3}3\theta -\cos 3\theta }$

• A. $tan\theta$
• B. $-tan\theta$
• C. $cot\theta$
• D. $-cot\theta$

Answer 1

The correct option is C $cot\theta$

We have $\frac{{\sin }^3\theta +\sin 3\theta }{{\cos }^{3}3\theta -\cos 3\theta }=\frac{{\sin }^3\theta +3\sin \theta -4{\sin }^3\theta }{{\cos }^{3}3\theta -(4{\cos }^3\theta -3\cos \theta )}$

$=\frac{3\sin \theta -3{\sin }^3\theta }{-3{\cos }^3\theta +3\cos \theta }$

$=\frac{\sin \theta (1-{\sin }^2\theta )}{\cos \theta (1-{\cos }^2\theta )}=\frac{\sin \theta {\cos }^2\theta }{{\sin }^2\theta \cos \theta }$

$=\frac{\cos \theta }{\sin \theta }$

$\frac{{\sin }^3\theta +\sin 3\theta }{{\cos }^{3}3\theta -\cos 3\theta }=\cot \theta$