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Question
Write the value of cos2 \[\left( \frac{1}{2} \cos^{- 1} \frac{3}{5} \right)\]
Solution
\[\text{Let }y = \cos^{- 1} \left( \frac{3}{5} \right)\]
\[ \Rightarrow \cos{y} = \frac{3}{5}\]
Now,
\[\cos^2 \left( \frac{1}{2} \cos^{- 1} \frac{3}{5} \right) = \cos^2 \left( \frac{1}{2}y \right)\]
\[ = \frac{\cos{y} + 1}{2} \left[ \because \cos2x = 2 \cos^2 x - 1 \right]\]
\[ = \frac{\frac{3}{5} + 1}{2}\]
\[ = \frac{\frac{8}{5}}{2}\]
\[ = \frac{4}{5}\]
∴ \[\cos^2 \left( \frac{1}{2} \cos^{- 1} \frac{3}{5} \right) = \frac{4}{5}\]
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