Question

If $tan \theta =3tan ⁡ \phi$ , then the maximum value of $\left(tan\right)^{2} \left(\theta - \phi\right)$ is (where, $tan \phi>0$ )

• A. $1$
• B. $\frac{1}{3}$
• C. $2$
• D. $4$

Answer 1

Answer: (B)

$tan \theta =3tan ⁡ \phi$
$tan \left(\theta - \phi\right)=\frac{tan ⁡ \theta - tan ⁡ \phi}{1 + tan ⁡ \theta tan ⁡ \phi}$
$=\frac{2 tan \phi}{1 + 3 tan^{2} ⁡ \phi}=\frac{2}{cot ⁡ \phi + 3 tan ⁡ \phi}$
Using $AM\geq GM$
$\frac{cot \phi + 3 tan ⁡ \phi}{2}\geq \sqrt{3}$
$\Rightarrow \, \frac{2}{cot \phi + 3 tan ⁡ \phi}\leq \frac{1}{\sqrt{3}}$
$\Rightarrow tan \left(\theta - \phi\right)\leq \frac{1}{\sqrt{3}}$
$\Rightarrow \tan ^{2}(\theta-\phi) \leq \frac{1}{3}$