Find $\cos (2 A)$, if $\cos (A)=-\frac{21}{29}$, and $A$ is in quadrant 3 . Write in fraction form

Step 1 ：We are given that \(\cos(A) = -\frac{21}{29}\) and that \(A\) is in the third quadrant.

Step 2 ：We know that the formula for \(\cos(2A)\) is \(2\cos^2(A) - 1\).

Step 3 ：Substitute \(\cos(A)\) into the formula to find \(\cos(2A)\).

Step 4 ：\(\cos(2A) = 2(-\frac{21}{29})^2 - 1\)

Step 5 ：Simplify to get \(\cos(2A) = \frac{41}{841}\)

Step 6 ：Final Answer: \(\boxed{\frac{41}{841}}\)