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

Question

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

Aicharya · Accepted Answer

$\begin{aligned} & 	an \left(2 	an ^{-1}\left(\frac{\sqrt{5}-1}{2}ight)ight) \ & =\frac{2 	an \left(	an ^{-1}\left(\frac{\sqrt{5}-1}{2}ight)ight)}{1-	an ^2\left(	an ^{-1}\left(\frac{\sqrt{5}-1}{2}ight)ight)} \ & =\frac{2\left(\frac{\sqrt{5}-1}{2}ight)}{1-\left(\frac{\sqrt{5}-1}{2}ight)^2} \ & =\frac{\sqrt{5}-1}{1-\left(\frac{6-2 \sqrt{5}}{4}ight)} \ & =\frac{4(\sqrt{5}-1)}{2 \sqrt{5}-2} \ & =\frac{4(\sqrt{5}-1)}{2(\sqrt{5}-1)}=2\end{aligned}$

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

Solution