The expression in the image is:

$\cos(2\pi - \alpha) \cdot \left(\tan\left(\frac{3\pi}{2} - \alpha\right)\right)^2 \cdot \tan(2\pi + \alpha) \cdot \left(\cot\left(\frac{\pi}{2} + \alpha\right)\right)^2$

Let's simplify it step by step:

1. Cosine of $2\pi - \alpha$: $\cos(2\pi - \alpha) = \cos(\alpha)$ because cosine is an even function.

2. Tangent of $\frac{3\pi}{2} - \alpha$: $\tan\left(\frac{3\pi}{2} - \alpha\right) = -\cot(\alpha)$ This follows from the tangent function's periodicity and transformations.

3. Tangent of $2\pi + \alpha$: $\tan(2\pi + \alpha) = \tan(\alpha)$ since the tangent function has a period of $\pi$.

4. Cotangent of $\frac{\pi}{2} + \alpha$: $\cot\left(\frac{\pi}{2} + \alpha\right) = -\tan(\alpha)$ because $\cot(\frac{\pi}{2} + x) = -\tan(x)$.

Substituting back into the original expression:

$\cos(\alpha) \cdot \left(-\cot(\alpha)\right)^2 \cdot \tan(\alpha) \cdot \left(-\tan(\alpha)\right)^2$

This simplifies to:

$\cos(\alpha) \cdot \cot^2(\alpha) \cdot \tan(\alpha) \cdot \tan^2(\alpha)$

Now, use the identity $\cot(\alpha) = \frac{1}{\tan(\alpha)}$:

$\cos(\alpha) \cdot \left(\frac{1}{\tan(\alpha)}\right)^2 \cdot \tan(\alpha) \cdot \tan^2(\alpha)$

Simplifying further:

$\cos(\alpha) \cdot \frac{1}{\tan^2(\alpha)} \cdot \tan(\alpha) \cdot \tan^2(\alpha)$

Cancel $\tan^2(\alpha)$ with the $\frac{1}{\tan^2(\alpha)}$:

$\cos(\alpha)$

Final simplified result:

The entire expression simplifies to $\cos(\alpha)$.

Would you like more details or have any questions?

Here are some related questions you can explore:

1. What are the key properties of trigonometric functions under transformations?
2. How does the periodicity of trigonometric functions affect their transformations?
3. Can cotangent and tangent identities help simplify other complex trigonometric expressions?
4. What is the relationship between even and odd functions, like cosine and sine?
5. How do you determine whether an expression can be simplified further?

Tip: When simplifying trigonometric expressions, always look for basic identities and function periodicity to reduce complexity.