Determine the value of tan2θ when sinθ=12/13 and pi/2 < θ < pi

Answer 1

Several ways to do this, this way came to mind
Did you recognize the 5-12-13 right-angled triangle?
So in quad II if sinθ = 12/13 , then cosθ = -5/13

tan 2θ = sin 2θ/cos 2θ
= 2sinθcosθ/(cos^2 θ - sin^2 θ)
= 2(12/13)(-5/13) / (25/169 - 144/169)
= (-120/169) / (-119/169)
= -120/-119
= 120/119

Answer 2

or, since you know that tanθ = -12/5 then
tan2θ = 2tanθ / (1 - tan^2θ)
= 2(-12/5) / (1 - (12/5)^2)
= 120/119