Answer
The value is $-\frac{7}{25}$.
Work Step by Step
In order to solve the value, apply the following steps given below:
$\theta ={{\tan }^{-1}}\left( -\frac{4}{3} \right)$
The value of tan is computed by dividing the perpendicular by the base. The perpendicular (y) is -4 and the base (x) is 3. By taking the help of the Pythagorian Theorem and computing the value of r, we get:
$\begin{align}
& {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\
& ={{\left( 3 \right)}^{2}}+{{\left( -4 \right)}^{2}} \\
& =9+16
\end{align}$
Further simplify the equation:
$\begin{align}
& r=\sqrt{25} \\
& =5
\end{align}$
So, the value of $\theta ={{\tan }^{-1}}\left( -\frac{4}{3} \right)$. So, it will become $\cos 2\theta $,
Solve it as follows:
$\begin{align}
& \cos \left[ 2{{\tan }^{-1}}\left( -\frac{4}{3} \right) \right]=\cos 2\theta \\
& =1-2{{\sin }^{2}}\theta \\
& =1-2{{\left( \frac{y}{r} \right)}^{2}} \\
& =1-2{{\left( \frac{-4}{5} \right)}^{2}}
\end{align}$
And solve the equation:
$\begin{align}
& \cos \left[ 2{{\tan }^{-1}}\left( -\frac{4}{3} \right) \right]=1-2\times \frac{16}{25} \\
& =-\frac{7}{25}
\end{align}$
So, the value will be $-\frac{7}{25}$.
Thus, the value of $\cos \left[ 2{{\tan }^{-1}}\left( -\frac{4}{3} \right) \right]$ is $-\frac{7}{25}$.
