Given $\theta = \frac{4}{\pi^3}$, we are to find the exact values of the following expressions:

(a) $\cos 2\theta$ (b) $2\cos \theta$ (c) $\cos \theta^2$

First, let's clarify each expression:

1. $\cos 2\theta$ refers to the cosine of twice the angle $\theta$.
2. $2 \cos \theta$ refers to twice the cosine of the angle $\theta$.
3. $\cos \theta^2$ refers to the cosine of the square of the angle $\theta$.

Given that the angle $\theta$ is provided in an unconventional form, it seems likely that these expressions are intended to be evaluated symbolically. Here’s how we proceed:

(a) $\cos 2\theta$

Using the double-angle formula for cosine: [ \cos