Let's analyze the given options one by one in relation to the law of cosines. The law of cosines states that for any triangle \( ABC \), the lengths of the sides (\( a \), \( b \), and \( c \)) and one of the angles (\( \theta \)) are related by the formula:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\][/tex]
This formula can also be written in terms of the other sides and angles:
[tex]\[
a^2 = b^2 + c^2 - 2bc \cos(A)
\][/tex]
[tex]\[
b^2 = a^2 + c^2 - 2ac \cos(B)
\][/tex]
Now let's examine the provided options:
1. Option A:
[tex]\[
b^2 = a^2 + c^2 - 2bc \cos(\ldots)
\][/tex]
This is incorrect because the term should involve \( -2ac \cos(B) \), not \( -2bc \cos \), so this does not match the law of cosines.
2. Option B:
[tex]\[
b^2 = a^2 - c^2 - 2bc \cos(C)
\][/tex]
This is incorrect because the term \( -c^2 \) should be \( +c^2 \), and the cosine term should involve angle \( B \), not \( C \).
3. Option C:
[tex]\[
b^2 = a^2 - c^2 - 2bc \cos(B)
\][/tex]
This is incorrect because similar to Option B, the term \( -c^2 \) is incorrect; it should be \( +c^2 \).
4. Option D:
[tex]\[
b^2 = a^2 + c^2 - 2ac \cos(B)
\][/tex]
This is correct. It matches the standard form of the law of cosines when solving for side \( b \) in terms of sides \( a \) and \( c \) and the angle \( B \).
Hence, after a detailed analysis, we conclude that Option D is indeed correctly representing the law of cosines.
So, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
