The numerical value of $ an \left(2 an ^{-1}\left(\frac{1}{5} ight)+\frac{\pi}{4} ight)$

Question

The numerical value of $	an \left(2 	an ^{-1}\left(\frac{1}{5}ight)+\frac{\pi}{4}ight)$

Aicharya · Accepted Answer

$\begin{array}{ll} & 	ext {Let } 2 	an ^{-1}\left(\frac{1}{5}ight)=x \ 	herefore \quad & 	an ^{-1}\left(\frac{1}{5}ight)=\frac{x}{2} \ 	herefore \quad & 	an \frac{x}{2}=\frac{1}{5} \ & 	ext { Using } \ & 	an 2 	heta=\frac{2 	an 	heta}{1-	an ^2 	heta} \ 	herefore \quad & 	an x=\frac{2 	an ^{\frac{x}{2}}}{1-	an ^2 \frac{x}{2}}\end{array}$ $\begin{aligned} & \Rightarrow 	an x=\frac{2 	imes \frac{1}{5}}{1-\frac{1}{25}} \ & \Rightarrow 	an x=\frac{5}{12}...(i) \end{aligned}$ $\begin{aligned} & 	ext { Consider } 	an \left(2 	an ^{-1}\left(\frac{1}{5}ight)+\frac{\pi}{4}ight) \ & =	an \left(x+\frac{\pi}{4}ight) \ & =\frac{	an x+	an \frac{\pi}{4}}{1-	an x \cdot 	an \frac{\pi}{4}} \ & =\frac{\frac{5}{12}+1}{1-\frac{5}{12}} \ & =\frac{\frac{17}{12}}{\frac{7}{12}}=\frac{17}{7}\end{aligned}$

The numerical value of $\tan \left(2 \tan ^{-1}\left(\frac{1}{5}\right)+\frac{\pi}{4}\right)$

Solution