Statement I In any triangle ABC, the square of the length of the bisector AD is `bc(1-(a^(2))/((b+c)^(2))).`
Statement II In any triangle ABC length of bisector AD is `(2bc)/((b+c))cos ((A)/(2)).`

Options

A. (a)Both Statement I and Statement II are correct and Statement II is the correct explanation of Statement I
B. (b)Both Statement I and Statement II are correct and Statement II is not the correct explanation of Statement I
C. (c)Statement I is correct but Statement II is incorrect
D. (d)Statement I is correct but Statement I is incorrect

Answer

To solve the problem, we need to verify the two statements regarding the angle bisector in triangle ABC.

### Step-by-Step Solution:

1. **Understanding the Angle Bisector**:
The angle bisector of a triangle divides the opposite side into two segments that are proportional to the adjacent sides. In triangle ABC, the angle bisector AD divides side BC into segments BD and DC.

2. **Statement I**:
We need to verify the formula for the square of the length of the angle bisector AD:
\[ AD^2 = bc \left(1 - \frac{a^2}{(b+c)^2}\right) \]
To derive this, we can use the formula for the length of the angle bisector:
\[ AD = \frac{2bc}{b+c} \cos\left(\frac{A}{2}\right) \]
Squaring both sides gives:
\[ AD^2 = \left(\frac{2bc}{b+c} \cos\left(\frac{A}{2}\right)\right)^2 = \frac{4b^2c^2 \cos^2\left(\frac{A}{2}\right)}{(b+c)^2} \]

3. **Using Cosine Half-Angle Identity**:
The cosine half-angle identity states:
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{1 + \cos A}{2}} \]
We know that:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting this into the cosine half-angle identity gives:
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{1 + \frac{b^2 + c^2 - a^2}{2bc}}{2}} = \sqrt{\frac{2bc + b^2 + c^2 - a^2}{4bc}} \]

4. **Substituting Back**:
Now substituting this back into the expression for \(AD^2\):
\[ AD^2 = \frac{4b^2c^2 \cdot \frac{2bc + b^2 + c^2 - a^2}{4bc}}{(b+c)^2} \]
Simplifying this, we get:
\[ AD^2 = \frac{bc(2bc + b^2 + c^2 - a^2)}{(b+c)^2} \]
This matches the form given in Statement I.

5. **Statement II**:
The length of the bisector is given by:
\[ AD = \frac{2bc}{b+c} \cos\left(\frac{A}{2}\right) \]
We have already derived this formula, confirming that Statement II is also true.

### Conclusion:
Both statements are true, and Statement II is indeed a correct explanation of Statement I.