Question

Write the value of cos⁻¹ $\left(\tan \frac{3\mathrm{\pi }}{4}\right)$.

Answer 1

We have
$$\begin{gathered} {\cos }^{-1}\left(\tan \frac{3\mathrm{\pi }}{4}\right)={\cos }^{-1}\left\{-\tan \left(\mathrm{\pi }-\frac{3\mathrm{\pi }}{4}\right)\right\}\left[\because \tan \left(\mathrm{\pi }-x\right)=-\tan x\right] \\ ={\cos }^{-1}\left\{\tan \left(-\frac{\mathrm{\pi }}{4}\right)\right\} \\ ={\cos }^{-1}\left\{-\tan \left(\frac{\mathrm{\pi }}{4}\right)\right\} \\ ={\cos }^{-1}\left(-1\right) \\ ={\cos }^{-1}\left(\mathrm{cos\pi }\right)\left[\because \mathrm{cos\pi }=-1\right] \\ =\mathrm{\pi } \end{gathered}$$

∴ ${\cos }^{-1}\left(\tan \frac{3\mathrm{\pi }}{4}\right)=\mathrm{\pi }$