Question

find the exact value of the given expression.

\\( \cos \left( 2 \tan ^ { - 1 } \left( \frac { 4 } { 3 } \
ight) \
ight) \\)

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:

$$
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$$

Answer:

\(-\frac{7}{25}\)