Find $\cos (2 A)$ , if $\cos (A) = - \frac{21}{29}$ , and $A$ is in quadrant 3
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So, the final answer is $0.04875148632580273$ .

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Step 1 ：We are given that $\cos (A) = - \frac{21}{29}$ and that $A$ is in quadrant 3.

Step 2 ：We are asked to find $\cos (2 A)$ .

Step 3 ：We know that the formula for $\cos (2 A)$ is $2 \cos^{2} (A) - 1$ .

Step 4 ：Substituting $\cos (A) = - \frac{21}{29}$ into the formula, we get $\cos (2 A) = 2 (- \frac{21}{29})^{2} - 1$ .

Step 5 ：Solving this, we find that $\cos (2 A) = 0.04875148632580273$ .

Step 6 ：Since $A$ is in quadrant 3, $\cos (2 A)$ will also be negative.

Step 7 ：So, the final answer is $0.04875148632580273$ .