The expression $1 + \cos(2\theta)$ can be simplified using trigonometric identities. One common approach is to use the double-angle identity for cosine:

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

Substitute this identity into $1 + \cos(2\theta)$:

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

Simplifying further:

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

Thus:

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

Alternatively, we can use the identity $\cos(2\theta) = 1 - 2\sin^2(\theta)$ and rewrite the expression in terms of $\sin(\theta)$:

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

Simplify:

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

Would you like more details, or do you have any questions?

Here are some related questions you might find helpful:

1. How does $\cos(2\theta)$ relate to other trigonometric identities?
2. How can we derive the double-angle identities for sine and cosine?
3. What other forms can $1 + \cos(2\theta)$ take using different trigonometric identities?
4. Can we simplify $1 + \cos^2(\theta)$ using similar methods?
5. How can $1 + \cos(2\theta)$ be applied in integration?

Tip: Recognizing key trigonometric identities, like the double-angle formulas, can simplify complex expressions and are especially useful in calculus and physics applications.