Find the exact value of \( \cos ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right) \) \( \cos ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right)=\square \) (Type an exact answer, using radicals as needed.)

To find the exact value of \( \cos^{2}\left(\frac{1}{2} \cos^{-1} \frac{3}{5}\right) \), let's follow these steps: 1. **Define the Angle:** Let \( \theta = \cos^{-1}\left(\frac{3}{5}\right) \). This means \( \cos \theta = \frac{3}{5} \). 2. **Use the Half-Angle Formula:** The half-angle formula for cosine is: \[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}} \] Since \( \theta \) is in the range \([0, \pi]\), \( \frac{\theta}{2} \) is in \([0, \frac{\pi}{2}]\), where cosine is positive. Therefore, we can ignore the negative square root. 3. **Square the Expression:** We need \( \cos^2\left(\frac{\theta}{2}\right) \): \[ \cos^2\left(\frac{\theta}{2}\right) = \left(\sqrt{\frac{1 + \cos \theta}{2}}\right)^2 = \frac{1 + \cos \theta}{2} \] 4. **Substitute \( \cos \theta \):** \[ \cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \frac{3}{5}}{2} = \frac{\frac{8}{5}}{2} = \frac{4}{5} \] **Final Answer:** \[ \cos^{2}\left(\frac{1}{2} \cos^{-1} \frac{3}{5}\right) = \frac{4}{5} \] **Answer:** \( \dfrac{4}{5} \)