To solve for \(\cos C\) using the Law of Cosines, we start with the given equation:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos C \][/tex]
We need to isolate \(\cos C\). Here are the steps to do so:
1. Start with the equation:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos C \][/tex]
2. Rearrange the equation to isolate the term involving \(\cos C\). Subtract \(a^2 + b^2\) from both sides:
[tex]\[ c^2 - a^2 - b^2 = -2ab \cos C \][/tex]
3. Divide both sides of the equation by \(-2ab\) to solve for \(\cos C\):
[tex]\[ \cos C = \frac{c^2 - a^2 - b^2}{-2ab} \][/tex]
4. Simplify the equation by factoring out a negative sign in the numerator:
[tex]\[ \cos C = \frac{-(a^2 + b^2 - c^2)}{2ab} \][/tex]
5. Rearranging the terms in the numerator gives:
[tex]\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]
Therefore, the correct expression for \(\cos C\) is:
[tex]\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]
Thus, the correct choice from the given options is:
[tex]\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]
