Mathematics

Problem

Let

f: (-1, 1) \to \mathbb{R}

be such that

f(\cos 4\theta) = \frac{2}{2 - \sec^2 \theta}

for

\theta \in \left( 0, \frac{\pi}{4} \right) \cup \left( \frac{\pi}{4}, \frac{\pi}{2} \right)

. Then, the value(s) of

f\left( \frac{1}{3} \right)

is/are a.

1 - \sqrt{\frac{3}{2}}

1 + \sqrt{\frac{3}{2}}

1 - \sqrt{\frac{2}{3}}

1 + \sqrt{\frac{2}{3}}