How find the exact value of cos(2Arctan (4/3))?

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Tony -This looks complicated, but if you take it one step at a time it will seem easy.  Start with Arctan(4/3).  We know that this equals an angle (let's call it θ).  Draw it on paper in the first quadrant.  Can you see that you have a 3-4-5 triangle? Great!Since tan(θ) = 4/3, and you know this is a 3-4-5 triangle, then we also know:sin(θ) = 4/5cos(θ) = 3/5Convince yourself these last two statements are true from your drawing!Now, use your trig identity for cos(2θ):cos(2θ) = [cos(θ)]^2 - [sin(θ)]^2Now plug in the values above and you get:cos(2θ) = [3/5]^2 - [4/5]^2cos(2θ) = (9-16)/25 = - 7/25ALWAYS DRAW A PICTURE!Hope that helps