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The cosine function is a fundamental concept in trigonometry, and it is especially important when you are dealing with angles in a right-angle triangle or the unit circle. The basic idea of the cosine function is to represent the adjacent side of an angle in a right triangle divided by the hypotenuse. This can be expressed as:

• \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)

On the unit circle, the cosine of an angle gives the x-coordinate of a point on the circle. The angle, represented usually by \( \theta \) or \( \alpha \), measures the counterclockwise rotation from the positive x-axis.
When you move to an angle in radians, cosine helps us understand the periodic nature of trigonometric functions. The function extends beyond what you might initially think of as simple triangles, applying to more complex geometric figures and even signal processing.

Half-angle formulas are a powerful tool in trigonometry. They allow you to find the trigonometric functions of half angles given the trigonometric function of the original angle. These formulas can transform complex expressions into simpler forms that are easier to work with.
The half-angle formula for cosine is:

• \( \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1+\cos\alpha}{2}} \)

The sign of the square root depends on the quadrant in which \( \frac{\alpha}{2} \) is located. These formulas are particularly handy because directly calculating the cosine of a specific angle can be a tedious process since it might not always yield a neat value.
Half-angle identities avoid unnecessary complications by working out these relationships in more straightforward terms.

Verifying equations in trigonometry is an important process that ensures your results are accurate. In the context of trigonometric identities, verifying means proving whether an equation holds true for all values or just for specific instances.
For the equation \( \cos \frac{\alpha}{2} = \frac{1}{2} \cos \alpha \), we investigate the problem by separately analyzing each side. At first glance, both sides might appear similar, but applying known trigonometric identities reveals discrepancies. It's critical to remember that an identity must hold true for any value of \( \alpha \), not just a selected few angles.
This equation, as revealed, does not satisfy that requirement and thus, is not an identity. Hence, verifying trigonometric equations demands a sharp eye for the nuances of mathematical operations and the transformations of these identities.