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

Question

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

Aicharya · Accepted Answer

$\begin{aligned} & 2 	an ^{-1}\left(\frac{1}{5}ight)-\frac{\pi}{4}=	an ^{-1} \frac{2 	imes \frac{1}{5}}{1-\left(\frac{1}{5}ight)^2}-	an ^{-1}(1)=	an ^{-1} \frac{5}{12}-	an ^{-1}(1) \ & =	an ^{-1}\left(\frac{\frac{5}{12}-1}{1+\frac{5}{12} 	imes 1}ight)=	an ^{-1}\left(\frac{-7}{17}ight) \ & \Rightarrow 	an \left\{2 	an ^{-1} \frac{1}{5}-\frac{\pi}{4}ight\}=\frac{-7}{17}\end{aligned}$

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

Solution