To find [tex]\(\cos(2x)\)[/tex] given that [tex]\(\sin(x) = -\frac{3}{5}\)[/tex] and [tex]\(\cos(x) < 0\)[/tex], let's follow these steps:
### Step 1: Determine [tex]\(\cot(x)\)[/tex]
Since [tex]\(\sin(x) = -\frac{3}{5}\)[/tex] and we need [tex]\(\cos(x)\)[/tex], we can use the Pythagorean identity:
[tex]\[
\sin^2(x) + \cos^2(x) = 1
\][/tex]
First, we calculate [tex]\(\sin^2(x)\)[/tex]:
[tex]\[
\sin^2(x) = \left( -\frac{3}{5} \right)^2 = \frac{9}{25}
\][/tex]
Substituting [tex]\(\sin^2(x)\)[/tex] into the Pythagorean identity:
[tex]\[
\frac{9}{25} + \cos^2(x) = 1
\][/tex]
Solving for [tex]\(\cos^2(x)\)[/tex]:
[tex]\[
\cos^2(x) = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}
\][/tex]
Since [tex]\(\cos(x) < 0\)[/tex], we take the negative square root:
[tex]\[
\cos(x) = -\sqrt{\frac{16}{25}} = -\frac{4}{5}
\][/tex]
### Step 2: Calculate [tex]\(\cos(2x)\)[/tex]
We use the double angle formula for cosine:
[tex]\[
\cos(2x) = \cos^2(x) - \sin^2(x)
\][/tex]
Substitute [tex]\(\cos(x)\)[/tex] and [tex]\(\sin(x)\)[/tex]:
[tex]\[
\cos^2(x) = \left( -\frac{4}{5} \right)^2 = \frac{16}{25}
\][/tex]
[tex]\[
\sin^2(x) = \left( -\frac{3}{5} \right)^2 = \frac{9}{25}
\][/tex]
Now substitute these values back into the double angle formula:
[tex]\[
\cos(2x) = \frac{16}{25} - \frac{9}{25} = \frac{16}{25} - \frac{9}{25} = \frac{7}{25}
\][/tex]
Thus, the value of [tex]\(\cos(2x)\)[/tex] is:
\[
\boxed{\frac{7}{25}}
