In a triangle ABC, the side c has two values , then `((a+b)^2)/(1+cosC)+((b-a)^2)/(1-cos C) = (2a^2)/(sin^2A)` True or false ?

Answer

To determine whether the statement

\[ \frac{(a+b)^2}{1+\cos C} + \frac{(b-a)^2}{1-\cos C} = \frac{2a^2}{\sin^2 A} \]

is true or false, we will analyze both sides step by step.

### Step 1: Analyze the Left-Hand Side (LHS)

The left-hand side of the equation is given by:

\[ LHS = \frac{(a+b)^2}{1+\cos C} + \frac{(b-a)^2}{1-\cos C} \]

To combine these two fractions, we need a common denominator. The common denominator will be \((1+\cos C)(1-\cos C) = 1 - \cos^2 C = \sin^2 C\).

### Step 2: Rewrite the LHS with the Common Denominator

Now, we can rewrite the LHS as:

\[ LHS = \frac{(a+b)^2(1-\cos C) + (b-a)^2(1+\cos C)}{\sin^2 C} \]

### Step 3: Expand the Numerator

Next, we expand the numerator:

1. Expand \((a+b)^2(1-\cos C)\):
\[ (a+b)^2(1-\cos C) = (a^2 + 2ab + b^2)(1 - \cos C) = a^2 + 2ab + b^2 - a^2\cos C - 2ab\cos C - b^2\cos C \]

2. Expand \((b-a)^2(1+\cos C)\):
\[ (b-a)^2(1+\cos C) = (b^2 - 2ab + a^2)(1 + \cos C) = b^2 - 2ab + a^2 + b^2\cos C - 2ab\cos C + a^2\cos C \]

### Step 4: Combine the Expanded Terms

Now, combine the expanded terms:

\[ LHS = \frac{(a^2 + 2ab + b^2 - a^2\cos C - 2ab\cos C - b^2\cos C) + (b^2 - 2ab + a^2 + b^2\cos C - 2ab\cos C + a^2\cos C)}{\sin^2 C} \]

Combine like terms:

\[ = \frac{2a^2 + 2b^2 + 0 - 4ab\cos C}{\sin^2 C} \]

### Step 5: Simplify the LHS

Thus, we have:

\[ LHS = \frac{2(a^2 + b^2 - 2ab\cos C)}{\sin^2 C} \]

### Step 6: Use the Cosine Rule

Using the cosine rule, we know that:

\[ c^2 = a^2 + b^2 - 2ab\cos C \]

So, we can rewrite the LHS as:

\[ LHS = \frac{2c^2}{\sin^2 C} \]

### Step 7: Analyze the Right-Hand Side (RHS)

The right-hand side is:

\[ RHS = \frac{2a^2}{\sin^2 A} \]

### Step 8: Establish the Relationship

For the equality \(LHS = RHS\) to hold, we need:

\[ \frac{2c^2}{\sin^2 C} = \frac{2a^2}{\sin^2 A} \]

This implies:

\[ \frac{c^2}{\sin^2 C} = \frac{a^2}{\sin^2 A} \]

### Step 9: Conclusion

This equality can hold true under certain conditions, particularly when the triangle has specific properties (like being isosceles or having certain angles). Therefore, the statement can be considered **True** under the condition that \(C\) has two values.