find the exact value of the given expression. \n\n \\( \\cos \\left( 2 \\tan ^ { - 1 } \\left( \\frac { 4 }…

Explanation:

Step1: Let (\theta=\tan^{-1}(\frac{4}{3}))

Then (\tan\theta = \frac{4}{3}). By the Pythagorean identity (\tan\theta=\frac{y}{x}), we can consider a right - triangle with (y = 4) and (x=3). Then, by the Pythagorean theorem (r=\sqrt{x^{2}+y^{2}}=\sqrt{3^{2}+4^{2}} = 5). So, (\cos\theta=\frac{3}{5})

Step2: Use the double - angle formula (\cos(2\alpha)=2\cos^{2}\alpha - 1)

Here (\alpha=\theta), so (\cos(2\tan^{-1}(\frac{4}{3}))=\cos(2\theta))

Substitute (\cos\theta=\frac{3}{5}) into the double - angle formula:

[ \begin{align*} \cos(2\theta)&=2\cos^{2}\theta - 1\ &=2\times(\frac{3}{5})^{2}-1\ &=2\times\frac{9}{25}-1\ &=\frac{18}{25}-1\ &=\frac{18 - 25}{25}\ &=-\frac{7}{25} \end{align*} ]

Answer:

(-\frac{7}{25})