Question

If $f\left(x\right)={\tan }^{-1}\left(2^x\right)+\lambda$ is an odd function, where $\lambda$ is

Answer 1

$f\left(x\right)$ is an odd function, so

$f(-x)=-f\left(x\right)$

${\tan }^{-1}\left(2^{-x}\right)+\lambda =-\left({\tan }^{-1}\right(2^x)+\lambda )$

${\tan }^{-1}\left(2^{-x}\right)+\lambda =-{\tan }^{-1}\left(2^x\right)-\lambda$

$-2\lambda ={\tan }^{-1}\left(2^{-x}\right)+{\tan }^{-1}\left(2^x\right)$

$-2\lambda ={\tan }^{-1}\left(\frac{1}{2^x}\right)+{\tan }^{-1}\left(2^x\right)$

$-2\lambda ={\cot }^{-1}\left(2^x\right)+{\tan }^{-1}\left(2^x\right)$ $\left({\cot }^{-1}\right(x)={\tan }^{-1}\left(\frac{1}{x}\right)forx>0)$

$-2\lambda =\frac{\pi }{2}$

$\lambda =-\frac{\pi }{4}$