Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

$\frac{\sin \theta }{\cos 3\theta }=\frac{2\sin \theta \cos \theta }{2\cos 3\theta \cos \theta }=\frac{\sin 2\theta }{2\cos 3\theta \cos \theta }=\frac{\sin (3\theta -\theta )}{2\cos 3\theta \cos \theta }$

$\therefore \frac{\sin \theta }{\cos 3\theta }=\frac{1}{2}[\tan 3\theta -\tan \theta ]\to (1)$

$\frac{\sin 3\theta }{\cos 9\theta }=\frac{1}{2}[\tan 9\theta -\tan 3\theta ]\to (2)$

$\frac{\sin 9\theta }{\cos 27\theta }=\frac{1}{2}[\tan 27\theta -\tan 9\theta ]\to (3)$

$\therefore (1)+(2)+(3)\Rightarrow \frac{1}{2}[\tan 27\theta -\tan \theta ]$ $\mathrm{A}=2,\mathrm{B}=27,\mathrm{C}=1$ $27A-B-27C=0$