Question

Write the value of tan⁻¹ $\left\{\tan \left(\frac{15\mathrm{\pi }}{4}\right)\right\}$.

Answer 1

We have
$$\begin{gathered} {\tan }^{-1}\left\{\tan \left(\frac{15\mathrm{\pi }}{4}\right)\right\}={\tan }^{-1}\left\{\tan \left(4\mathrm{\pi }-\frac{\mathrm{\pi }}{4}\right)\right\} \\ {\tan }^{-1}\left\{-\tan \left(\frac{\mathrm{\pi }}{4}\right)\right\}\left[\because \tan \left(4\mathrm{\pi }-x\right)=-\tan x\right] \\ ={\tan }^{-1}\left\{\tan \left(-\frac{\mathrm{\pi }}{4}\right)\right\} \\ =-\frac{\mathrm{\pi }}{4}\left[\because {\tan }^{-1}\left(\tan x\right)=x\right] \end{gathered}$$


∴ ${\tan }^{-1}\left\{\tan \left(\frac{15\pi }{4}\right)\right\}=-\frac{\pi }{4}$