Question
Find the value of:
\[ \cos^2\left(\frac{1}{2}\cos^{-1}\left(\frac{3}{5}\right)\right) \]
Solution
Let
\[ \theta = \cos^{-1}\left(\frac{3}{5}\right) \]
Then,
\[ \cos \theta = \frac{3}{5} \]
Using identity:
\[ \cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos \theta}{2} \]
Substitute:
\[ \cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \frac{3}{5}}{2} \]
\[ = \frac{\frac{8}{5}}{2} = \frac{8}{10} = \frac{4}{5} \]
Final Answer:
\[ \boxed{\frac{4}{5}} \]
Key Concept
Use the identity \( \cos^2(\theta/2) = \frac{1 + \cos\theta}{2} \) for quick simplification.