Simplify :
`(x^(2)-(y-z)^(2))/((x+z)^(2)-y^(2))+(y^(2)-(x-z)^(2))/((x+y)^(2)-z^(2))+(z^(2)-(x-y)^(2))/((y+z)^(2)-x^(2))`

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To simplify the given expression

\[ \frac{x^2 - (y - z)^2}{(x + z)^2 - y^2} + \frac{y^2 - (x - z)^2}{(x + y)^2 - z^2} + \frac{z^2 - (x - y)^2}{(y + z)^2 - x^2} \]

we will use the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\).

### Step 1: Simplify the first term

The first term is

\[ \frac{x^2 - (y - z)^2}{(x + z)^2 - y^2} \]

Using the difference of squares formula:

- Numerator:
\[ x^2 - (y - z)^2 = (x + (y - z))(x - (y - z)) = (x + y - z)(x - y + z) \]

- Denominator:
\[ (x + z)^2 - y^2 = ((x + z) + y)((x + z) - y) = (x + z + y)(x + z - y) \]

Thus, the first term simplifies to:

\[ \frac{(x + y - z)(x - y + z)}{(x + z + y)(x + z - y)} \]

### Step 2: Simplify the second term

The second term is

\[ \frac{y^2 - (x - z)^2}{(x + y)^2 - z^2} \]

Using the difference of squares formula:

- Numerator:
\[ y^2 - (x - z)^2 = (y + (x - z))(y - (x - z)) = (y + x - z)(y - x + z) \]

- Denominator:
\[ (x + y)^2 - z^2 = ((x + y) + z)((x + y) - z) = (x + y + z)(x + y - z) \]

Thus, the second term simplifies to:

\[ \frac{(y + x - z)(y - x + z)}{(x + y + z)(x + y - z)} \]

### Step 3: Simplify the third term

The third term is

\[ \frac{z^2 - (x - y)^2}{(y + z)^2 - x^2} \]

Using the difference of squares formula:

- Numerator:
\[ z^2 - (x - y)^2 = (z + (x - y))(z - (x - y)) = (z + x - y)(z - x + y) \]

- Denominator:
\[ (y + z)^2 - x^2 = ((y + z) + x)((y + z) - x) = (y + z + x)(y + z - x) \]

Thus, the third term simplifies to:

\[ \frac{(z + x - y)(z - x + y)}{(y + z + x)(y + z - x)} \]

### Step 4: Combine all terms

Now we have:

\[ \frac{(x + y - z)(x - y + z)}{(x + z + y)(x + z - y)} + \frac{(y + x - z)(y - x + z)}{(x + y + z)(x + y - z)} + \frac{(z + x - y)(z - x + y)}{(y + z + x)(y + z - x)} \]

### Step 5: Cancel common terms

Notice that in each term, the denominators have common factors that can cancel out. After simplification, we will find that all terms in the numerator will combine to yield \(x + y + z\).

### Final Result

Thus, the entire expression simplifies to:

\[ \frac{x + y + z}{x + y + z} = 1 \]