Question

Given that π < 𝑥 < 2π and tan 𝑥 = 3, determine the exact value of cos(2𝑥).

Answer 1

in QIII, draw a standard triangle with legs 3 and 1. It should be clear that
cosθ = x/r = -3/√10
cos(2x) = 2cos^2(x) - 1 = 2 * 9/10 - 1 = 4/5

Answer 2

should be -4/5

should have been:
cosθ = x/r = -1/√10
cos(2x) = 2cos^2(x) - 1 = 2 * 1/10 - 1 = - 4/5

(caught your typo by doing cos (2x) = cos^2 x - sin^2 x)