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 \( \cos^{2}\left(\frac{1}{2} \cos^{-1} \frac{3}{5}\right) \), we can use the half-angle identity for cosine: \[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} \] In this case, let \( \theta = \cos^{-1} \frac{3}{5} \). From this, we know that: \[ \cos(\theta) = \frac{3}{5} \] Now apply the half-angle identity: \[ \cos\left(\frac{1}{2} \theta\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} = \sqrt{\frac{1 + \frac{3}{5}}{2}} = \sqrt{\frac{\frac{5}{5} + \frac{3}{5}}{2}} = \sqrt{\frac{\frac{8}{5}}{2}} = \sqrt{\frac{8}{10}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \] Now, since we need \( \cos^{2}\left(\frac{1}{2} \theta\right) \): \[ \cos^{2}\left(\frac{1}{2} \theta\right) = \left( \frac{2}{\sqrt{5}} \right)^{2} = \frac{4}{5} \] Thus, the exact value is: \[ \cos^{2}\left(\frac{1}{2} \cos^{-1} \frac{3}{5}\right) = \frac{4}{5} \] Therefore, the answer is \[ \boxed{\frac{4}{5}} \]