Question

$\tan \left(2{\tan }^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)=$

• A. $4$
• B. $3$
• C. $2$
• D. $6$

Answer 1

The correct option is C $2$

The value of $\tan \left(2{\tan }^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)$ is

$=\frac{2\tan \left({\tan }^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)}{1-{\tan }^2\left({\tan }^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)}$

$=\frac{\sqrt{5}-1}{1-{\left(\frac{\sqrt{5}-1}{2}\right)}^2}$

$=\frac{4(\sqrt{5}-1)}{4-(5+1-2\sqrt{5})}$

$=\frac{4(\sqrt{5}-1)}{2\sqrt{5}-2}$

$=\frac{4(\sqrt{5}-1)}{2(\sqrt{5}-1)}$

$=2$