The expression $2 \cos^2 \left( \frac{x}{2} \right)$ doesn't simplify into a more elementary form by itself, but we can express it in terms of trigonometric identities.

One well-known identity for trigonometric functions is the double-angle identity for cosine:

$\cos(2\theta) = 2 \cos^2(\theta) - 1$

If we use this identity for $\theta = \frac{x}{2}$, we can rewrite $\cos^2 \left( \frac{x}{2} \right)$:

$\cos^2 \left( \frac{x}{2} \right) = \frac{1 + \cos(x)}{2}$

Now, we can substitute this back into the original expression:

$2 \cos^2 \left( \frac{x}{2} \right) = 2 \times \frac{1 + \cos(x)}{2} = 1 + \cos(x)$

So, the simplified form of $2 \cos^2 \left( \frac{x}{2} \right)$ is:

$1 + \cos(x)$

Would you like further details on trigonometric identities or have any questions on this? Here are 5 related questions:

1. What are other important trigonometric identities?
2. How does the cosine double angle formula derive from the Pythagorean identity?
3. How do you apply the half-angle formulas for trigonometric functions?
4. How can this be applied to solving trigonometric equations?
5. What is the relationship between sine and cosine in trigonometric identities?

Tip: Always try to recognize familiar identities to simplify trigonometric expressions quickly!