Question

use the following information to determine cos(2x). tan(x) = 2/3 and cos(x) is positive

Explanation:

Step1: Recall the identity $\tan(x)=\frac{\sin(x)}{\cos(x)}$

Given $\tan(x)=\frac{2}{3}=\frac{\sin(x)}{\cos(x)}$, so $\sin(x)=\frac{2}{3}\cos(x)$.

Step2: Use the Pythagorean identity $\sin^{2}(x)+\cos^{2}(x) = 1$

Substitute $\sin(x)=\frac{2}{3}\cos(x)$ into $\sin^{2}(x)+\cos^{2}(x)=1$. We get $(\frac{2}{3}\cos(x))^{2}+\cos^{2}(x)=1$.
Expanding gives $\frac{4}{9}\cos^{2}(x)+\cos^{2}(x)=1$.
Combining like - terms: $(\frac{4}{9} + 1)\cos^{2}(x)=1$, so $\frac{13}{9}\cos^{2}(x)=1$.
Then $\cos^{2}(x)=\frac{9}{13}$, and since $\cos(x)>0$, $\cos(x)=\frac{3}{\sqrt{13}}$.

Step3: Recall the double - angle formula for cosine

The double - angle formula for cosine is $\cos(2x)=2\cos^{2}(x)-1$.

Step4: Substitute the value of $\cos^{2}(x)$ into the double - angle formula

Substitute $\cos^{2}(x)=\frac{9}{13}$ into $\cos(2x)=2\cos^{2}(x)-1$.
We have $\cos(2x)=2\times\frac{9}{13}-1=\frac{18}{13}-1=\frac{18 - 13}{13}=\frac{5}{13}$.

Answer:

$\frac{5}{13}$