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., $\sin 2\theta =1$
$\Rightarrow 2\theta =\frac{\pi }{2}$
$\therefore \theta =\frac{\pi }{4}$