1. (c2+a2)/(b2+c2)={1+cosBcos(C-A)}/{1+cosAcos(B-c)}
2. (a+b+c)[tanA/2+tanB/2]=2c cotC/2
3. sin(A-B)/2=(a-b)/c.cosC/2
4. b.cosb + c.cosc = a.cos(b-c)
5. a cosA+b cosB=c cos(A-B)
6. b2=(c-a)2.cos2(B/2)+(c+a)2.sin2(B/2)

It looks like you're diving into some complex trigonometric identities and relationships involving angles and sides of a triangle. Let's break down these equations step by step to understand their significance and how they relate to each other.

Understanding the Triangle Relationships

In trigonometry, particularly in the context of triangles, we often deal with relationships between the sides and angles. The equations you've provided seem to involve various trigonometric identities that can be derived from the Law of Cosines and the Law of Sines. Let's analyze them one by one.

Breaking Down the First Equation

The first equation you mentioned is:

(c² + a²) / (b² + c²) = (1 + cosB * cos(C - A)) / (1 + cosA * cos(B - C))

This equation relates the sides of the triangle to the cosines of its angles. The left side represents a ratio of the squares of the sides, while the right side involves cosine functions. This can be useful in proving various properties of triangles, especially when you want to express one side in terms of the others and their angles.

Exploring the Second Equation

The next equation is:

(a + b + c) * [tan(A/2) + tan(B/2)] = 2c * cot(C/2)

This equation connects the semi-perimeter of the triangle (a + b + c) with the tangents of half the angles. The term cot(C/2) is particularly interesting because it relates to the triangle's area and can be derived from the triangle's circumradius. This relationship is often used in advanced geometry to find unknown sides or angles.

Analyzing the Third Equation

Next, we have:

sin(A - B) / 2 = (a - b) / (c * cos(C/2))

This equation expresses the sine of the difference of two angles in terms of the sides of the triangle. It highlights how the difference in lengths of two sides can affect the angle between them. This is particularly useful in solving for angles when the sides are known.

Delving into the Fourth Equation

Now, let's look at:

b² = (c - a)² * cos²(B/2) + (c + a)² * sin²(B/2)

This equation is a form of the Law of Cosines, expressed in terms of half-angles. It shows how the square of one side can be expressed as a combination of the other two sides, weighted by the cosine and sine of half of angle B. This can be particularly useful in problems involving angle bisectors or when working with triangle area calculations.

Final Thoughts on the Last Equation

The last equation:

a * cosA + b * cosB = c * cos(A - B)

This relationship is a vector-based approach to understanding triangle sides and angles. It shows how the projections of the sides onto the direction of one of the angles can be equated to the cosine of the angle difference. This is particularly useful in physics and engineering applications where forces are resolved into components.

Conclusion

Each of these equations provides a unique insight into the relationships within a triangle. They can be used to derive unknown values or to prove other geometric properties. Understanding these relationships is crucial for solving complex problems in trigonometry and geometry. If you have specific areas within these equations that you'd like to explore further, feel free to ask!