Question

Consider the following statements
Statement I If $3 \tan \left(\theta-15^{\circ}\right)=\tan \left(\theta+15^{\circ}\right)$
$0^{\circ} < \theta < 90^{\circ}$, then $\theta=\frac{\pi}{4}$
Statement II $\tan 3 \theta=\frac{3 \tan \theta-\tan ^3 \theta}{1-3 \tan ^2 \theta}$
Choose the correct option.

• A. Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I
• B. Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I
• C. Statement I is true; Statement II is false
• D. Statement I is false; Statement II is true

Answer 1

Answer: (B)

Given that, $3 \tan \left(\theta-15^{\circ}\right)=\tan \left(\theta+15^{\circ}\right)$ which can be rewritten as
$\frac{\tan \left(\theta+15^{\circ}\right)}{\tan \left(\theta-15^{\circ}\right)}=\frac{3}{1}$
Applying componendo and dividendo, we get
$\frac{\tan \left(\theta+15^{\circ}\right)+\tan \left(\theta-15^{\circ}\right)}{\tan \left(\theta+15^{\circ}\right)-\tan \left(\theta-15^{\circ}\right)}=2$
$\frac{\sin \left(\theta+15^{\circ}\right) \cos \left(\theta-15^{\circ}\right)+\sin \left(\theta-15^{\circ}\right) \cos \left(\theta+15^{\circ}\right)}{\sin \left(\theta+15^{\circ}\right) \cos \left(\theta-15^{\circ}\right)-\sin \left(\theta-15^{\circ}\right) \cos \left(\theta-15^{\circ}\right)}=2$
$\Rightarrow \frac{\sin 2 \theta}{\sin 30^{\circ}}=2$ i.e., $\sin 2 \theta=1$
$\Rightarrow 2 \theta=\frac{\pi}{2}$
$\therefore \theta=\frac{\pi}{4}$