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.)

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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:** $$ \dfrac{4}{5} $$
$$ \cos^{2}\left(\frac{1}{2} \cos^{-1} \frac{3}{5}\right) = \frac{4}{5} $$

Find the exact value of

(Type an exact answer, using radicals as needed.)

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