Find the exact value, if any, of the following composite function. Do not use a calculator. \( \cos ^{-1}\left[\cos \left(\frac{7 \pi}{10}\right)\right] \) Select the correct choice below and, if necessary, fill in the answer box within your choice. A. \( \cos ^{-1}\left[\cos \left(\frac{7 \pi}{10}\right)\right]= \) (Simplify your answer. Type an exact answer, using \( \pi \) as needed. Use integers or fractions for any numbers in the expression.) B. It is not defined.

To find the value of \( \cos^{-1}\left[\cos\left(\frac{7\pi}{10}\right)\right] \), we need to understand the properties of the inverse cosine function. The function \( \cos^{-1}(x) \) returns values in the range \( [0, \pi] \). The cosine function is periodic and symmetric, which means that for angles outside of this range, we may need to adjust the angle to find the corresponding value within the range of the inverse function. 1. First, we calculate \( \cos\left(\frac{7\pi}{10}\right) \). The angle \( \frac{7\pi}{10} \) is in the second quadrant, where cosine is negative. 2. The cosine function is symmetric about \( \pi \). Therefore, we can find an equivalent angle in the first quadrant: \[ \frac{7\pi}{10} = \pi - \frac{3\pi}{10} \] This means: \[ \cos\left(\frac{7\pi}{10}\right) = -\cos\left(\frac{3\pi}{10}\right) \] 3. Now, we apply the inverse cosine function: \[ \cos^{-1}\left[\cos\left(\frac{7\pi}{10}\right)\right] \] Since \( \frac{7\pi}{10} \) is in the second quadrant, we need to find the equivalent angle in the range \( [0, \pi] \). The angle \( \frac{7\pi}{10} \) itself is already in this range. Thus, we have: \[ \cos^{-1}\left[\cos\left(\frac{7\pi}{10}\right)\right] = \frac{7\pi}{10} \] Therefore, the exact value is: \[ \boxed{\frac{7\pi}{10}} \]