If the angles \(\mathrm{A}<\mathrm{B}<\mathrm{C}\) of a \(\triangle \mathrm{ABC}\) are in \(\mathrm{A} . \mathrm{P}\). then (A) \(c^{2}=a^{2}+b^{2}-a b\) (B) \(c^{2}=a^{2}+b^{2}\) (C) \(b^{2}=a^{2}+c^{2}-a c\) (D) \(a^{2}=b^{2}+c^{2}-b c\)

An arithmetical progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. This constant difference is known as the common difference of the AP. In the context of triangles, when the angles of a triangle are in arithmetical progression, there exists a relationship between those angles.

Considering the angles of a triangle, if we denote them as A, B, and C, and they are in AP, then angle B is the average of angles A and C. This implies that the difference between angles B and A is the same as the difference between angles C and B. The significance of such an arrangement allows us to further explore and establish connections between the angles and sides of the triangle using trigonometric concepts like the law of cosines.

In any triangle, the sum of the three internal angles is always equal to 180 degrees. This fundamental property leads to various relationships that can be established within a triangle. When the angles are in arithmetical progression, there is a specific way to relate them, as we saw with the formula for the middle angle, B, being the average of the other two, A and C.

The angle relationships play a critical role when applying trigonometric rules, such as the law of cosines, which requires an understanding of how each angle opposes a specific side in a triangle. Knowing these relationships allows us to manipulate and express one angle in terms of the others, thus simplifying complex expressions and, ultimately, solving for unknown sides or angles in a triangle.

The cosine double angle formula is a trigonometric identity that expresses the cosine of twice an angle in terms of the sine or cosine of the original angle. The standard forms of the cosine double angle formula are:

• \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\)
• \(\cos(2\theta) = 2\cos^2(\theta) - 1\)
• \(\cos(2\theta) = 1 - 2\sin^2(\theta)\)

In the context of a triangle, this formula is particularly useful when trying to relate the sides of a triangle to its angles. By applying it, as seen in the step-by-step solution, we transform the expression for angle C into a form that can be directly related to angles A and B. This transformation paves the way for utilizing the law of cosines to find a relationship between the triangle's sides that is reflective of its internal angles' progression.