If $\tan A = \frac{4}{3}$ then the value of $\cos 2A$ is
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Always double-check your signs! Since $\tan A = 4/3$ (greater than 1), $A$ is greater than $45^\circ$, meaning $2A$ is greater than $90^\circ$. In the second quadrant, cosine must be negative, immediately eliminating options (C) and (D).
2. Substitution and Calculation: Given $\tan A = \frac{4}{3}$, we first calculate $\tan^2 A$:
$$\tan^2 A = \left(\frac{4}{3}\right)^2 = \frac{16}{9}$$
Now substitute this into the formula:
$$\cos 2A = \frac{1 - \frac{16}{9}}{1 + \frac{16}{9}}$$
Simplify the numerator and denominator by using a common denominator of 9:
$$\cos 2A = \frac{\frac{9 - 16}{9}}{\frac{9 + 16}{9}}$$
$$\cos 2A = \frac{-7/9}{25/9}$$
$$\cos 2A = -\frac{7}{25}$$
The negative result indicates that $2A$ is in an quadrant where cosine is negative (the second quadrant).