Pls explain this?

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{It} \mathrm{is} \mathrm{solved} \mathrm{incorrectly} \mathrm{in} \mathrm{your} \mathrm{book}. \\ \mathrm{Becuase} {\tan }^{-1} \mathrm{ranges} \mathrm{from} -\frac{\mathrm{\pi }}{2} \mathrm{to} \frac{\mathrm{\pi }}{4} \mathrm{and} \mathrm{cannot} \mathrm{give} \frac{3\mathrm{\pi }}{4} \\ \mathrm{We} \mathrm{have} \\ {\tan }^{-1}\left(-\tan \frac{\mathrm{\pi }}{4}\right) \\ =-{\tan }^{-1}\left(\tan \frac{\mathrm{\pi }}{4}\right) \left[\mathrm{As} {\tan }^{-1}\left(-\mathrm{A}\right)=-\tan \left(\mathrm{A}\right)\right] \\ =-\frac{\mathrm{\pi }}{4} \mathrm{which} \mathrm{is} \mathrm{correct}. \end{gathered}$$

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