Question

Find the exact value of the given expression.

cos(2 tan^-1 12/5)

Answer 1

you need two things. "double angle formula" and also
\[\cos(\tan^{-1}(\frac{12}{5}))\]

Answer 2

from the picture we see that
\[\cos(\tan^{-1}(\frac{12}{5})=\frac{5}{13}\] and so now this is like finding
\[\cos(2\theta)\] knowing that
\[\cos(\theta)=\frac{5}{13}\]
\[\cos(2\theta)=2\cos^2(\theta)-1\] so use
\[\cos(2\theta)=2\times (\frac{5}{13})^2-1\] to find your answer

Answer 3

From Mathematica:\[\text{TrigFactor}\left( \cos \left(2 \text{ }\tan ^{-1}\left(\frac{12}{5}\right)\right) \right) = -\frac{119}{169} \]The LHS and the RHS converted to a decimal number respectively:\[\text{Cos}\left[2 \text{ ArcTan}\left[\frac{12}{5}\right]\right]\text{ //}N \to -0.704142 \]\[-\frac{119}{169}\text{ //}N\to -0.704142 \]