Find the exact value of the expression. Do not use a calculator.
-$$\tan ^ { - 1 } \left[ \tan \left( - \frac { \pi } { 8 } \right) \right]$$
A) $- 8 \pi$
B) $\frac { \pi } { 8 }$
C) $8 \pi$
D) $- \frac { \pi } { 8 }$

Question

Quizplus · Accepted Answer

The function $\tan^{-1}(\tan(x))$ essentially finds the angle whose tangent is $\tan(x)$. Since the tangent function is periodic with a period of $\pi$, $\tan^{-1}(\tan(x))$ will return a value in the principal branch, which is $-\frac{\pi}{2} < x < \frac{\pi}{2}$. For $x = -\frac{\pi}{8}$, which is within the principal branch, $\tan^{-1}(\tan(-\frac{\pi}{8})) = -\frac{\pi}{8}$.

Find the Exact Value of the Expression tan⁡−1[tan⁡(−π8)]\tan ^ { - 1 } \left[ \tan \left( - \frac { \pi } { 8 } \right) \right]tan−1[tan(−8π​)]

Find the Exact Value of the Expression $\tan ^ { - 1 } \left[ \tan \left( - \frac { \pi } { 8 } \right) \right]$