Find the exact value of cos(2 arctan 4/3) without a calculator? answer with details and explanations if you can.. i'm really confused and there's a test tomorrow. : (

Let θ = arc tan 4/3, then
tan θ = 4/3
The scenario describes a right angled triangle with angle θ and the side opposite to θ is 4 and the side adjacent to θ is 3 and the hypotenus is 5.
Thus cos θ = adj/hyp = 3/5 and sin θ = opp/hyp = 4/5

Now, cos (2 arctan 4/3) = cos 2θ = cos^2 θ - sin^2 θ = (3/5)^2 - (4/5)^2 = 9/25 - 16/25 = -7/25

Therefore, cos (2 arctan 4/3) = -7/25