Question

Write the value of cos⁻¹ \[\left( \tan\frac{3\pi}{4} \right)\]

Answer 1

We have

\[\cos^{- 1} \left( \tan\frac{3\pi}{4} \right) = \cos^{- 1} \left\{ - \tan\left( \pi - \frac{3\pi}{4} \right) \right\} \left[ \because \tan\left( \pi - x \right) = - \tan{x} \right]\]
\[ = \cos^{- 1} \left\{ \tan\left( - \frac{\pi}{4} \right) \right\}\]
\[ = \cos^{- 1} \left\{ - \tan\left( \frac{\pi}{4} \right) \right\}\]
\[ = \cos^{- 1} \left( - 1 \right)\]
\[ = \cos^{- 1} \left( cos\pi \right) \left[ \because cos\pi = - 1 \right]\]
\[ = \pi\]

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