Question

If $\theta$ and $\varphi$ are positive angles such that $\theta +\varphi =\frac{\pi }{4}$, then maximum value of $\tan \theta \tan \varphi$ is

• A. $\sqrt{2}+1$
• B. $\sqrt{2}-1$
• C. $\frac{\sqrt{2}+1}{\sqrt{2}-1}$
• D. $\frac{\sqrt{2}-1}{\sqrt{2}+1}$

Answer 1

The correct option is A $\sqrt{2}+1$

We have

$\theta$ and $\varphi$ are postive angles such that $\theta +\varphi =\frac{\pi }{4}$

Now, the maximum value of $\tan \theta \tan \varphi$

$\Rightarrow \theta =\left(\frac{\pi }{4}-\varphi \right)$

$\Rightarrow \tan \theta =\tan \left(\frac{\pi }{4}-\varphi \right)$

$=\sqrt{2}+1$

Hence, the option $\left(A\right)$ is correct.