In `DeltaABC,(a-b)^(2)cos^(2).(C)/(2)+(a+b)^(2)sin^(2).(C)/(2)`=

Options

A. `b^(2)`
B. `c^(2)`
C. `a^(2)`
D. `a^(2)+b^(2)+c^(2)`

Answer

To simplify the expression \((a-b)^2 \cos^2\left(\frac{C}{2}\right) + (a+b)^2 \sin^2\left(\frac{C}{2}\right)\), we will follow these steps:

### Step 1: Expand the squares
We start by expanding \((a-b)^2\) and \((a+b)^2\):

\[ (a-b)^2 = a^2 - 2ab + b^2 \]
\[ (a+b)^2 = a^2 + 2ab + b^2 \]

### Step 2: Substitute the expansions into the expression
Now, substitute these expansions into the original expression:

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

### Step 3: Distribute the trigonometric functions
Distributing \(\cos^2\left(\frac{C}{2}\right)\) and \(\sin^2\left(\frac{C}{2}\right)\):

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

### Step 4: Combine like terms
Now, we can combine the terms:

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

Using the identity \(\cos^2 x + \sin^2 x = 1\):

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

### Step 5: Simplify further
Recall that \(\sin^2\left(\frac{C}{2}\right) - \cos^2\left(\frac{C}{2}\right) = -\cos C\):

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

### Step 6: Recognize the final form
The expression simplifies to:

\[ = (a-b)^2 + c^2 \]

### Final Result
Thus, the simplified expression is:

\[ = c^2 \]