Please provide the mathematical expression shown in the image.

The mathematical expressions shown in the image are:

1. \( \frac{\left(1 - 8 \cos^2 \frac{\alpha}{2} \sin^2 \frac{\alpha}{2}\right) \left(2 \cos^2 \frac{\alpha}{2} - 1\right)}{\cos 3\alpha + \cos \alpha} \)
2. \( \left(\frac{\sin \alpha}{\sin 2\alpha} - \frac{\cos \alpha}{\cos 2\alpha}\right) \cdot \frac{\cos \alpha - \cos 7\alpha}{\sin \alpha} \)
3. \( \left(\frac{\sin 4\alpha}{\sin \alpha} - \frac{\cos 4\alpha}{\cos \alpha}\right) \left(\frac{1}{\sin 3\alpha} + \frac{1}{\sin \alpha}\right) \)
4. \( \cos^4 \alpha = \frac{3}{8} + \frac{1}{2} \cos 2\alpha + \frac{1}{8} \cos 4\alpha \)