Question

Write the value of tan⁻¹\[\left\{ \tan\left( \frac{15\pi}{4} \right) \right\}\]

Answer 1

We have
\[\tan^{- 1} \left\{ \tan\left( \frac{15\pi}{4} \right) \right\} = \tan^{- 1} \left\{ \tan\left( 4\pi - \frac{\pi}{4} \right) \right\}\]
\[ \tan^{- 1} \left\{ - \tan\left( \frac{\pi}{4} \right) \right\} \left[ \because \tan\left( 4\pi - x \right) = - \tan{x} \right]\]
\[ = \tan^{- 1} \left\{ \tan\left( - \frac{\pi}{4} \right) \right\} \]
\[ = - \frac{\pi}{4} \left[ \because \tan^{- 1} \left( \tan{x} \right) = x \right] \]
∴ \[\tan^{- 1} \left\{ \tan\left( \frac{15\pi}{4} \right) \right\} = - \frac{\pi}{4}\]