How to solve "If $\tan x = 24/7$ and $x$ is in the first quadrant, then find $\cos(2x)$."

Question

One of my problems for homework is "If $\tan(x)=24/7$ and $x$ is in the first quadrant then find $\cos(2x)$." I keep getting to $\cos(7/25 + 7/25)$ which evaluates to $.84726$, which isn't an option. My options are:

A- $-\dfrac{527}{625}$

B- $1$

C- $-\dfrac{527}{81}$

D- $\dfrac{14}{25}$

Any help would be appreciated, thanks.

PS. None of the fractions equal what I got either.

Answer 1

$\tan x = \frac{24}{7}$

From considering the special $7,24,25$ right triangle, and the fact that $x$ is acute, you immediately get $\sin x = \frac{24}{25}, \cos x= \frac{7}{25}$.

$\cos 2x = 2\cos^2x - 1 = 2(\frac{7}{25})^2 - 1 = -\frac{527}{625}$

Answer 2

Hint:

Use $$\cos2x=\dfrac{1-\tan^2x}{1+\tan^2x}$$