There is a property of the tantan function that states:
if tan(x/2) = ttan(x2)=t then
sin(x) = (2t)/(1+t^2)sin(x)=2t1+t2
From here you write the equation
(2t)/(1+t^2) = 3/52t1+t2=35
rarr 5*2t = 3(1+t^2)→5⋅2t=3(1+t2)
rarr 10t = 3t^2+3→10t=3t2+3
rarr 3t^2-10t+3 = 0→3t2−10t+3=0
Now you find the roots of this equation:
Delta = (-10)^2 - 4*3*3 = 100-36 = 64
t_(-) = (10-sqrt(64))/6 = (10-8)/6 = 2/6 = 1/3
t_(+) = (10+sqrt(64))/6 = (10+8)/6 = 18/6 = 3
Finaly you have to find which of the above answers is the right one. Here is how you do it:
Knowing that 90°< x <180° then 45°< x/2 <90°
Knowing that on this domain, cos(x) is a decreasing function and sin(x) is an increasing function, and that sin(45°) = cos(45°)
then sin(x/2) > cos(x/2)
Knowing that tan(x) = sin(x)/cos(x) then in our case tan(x/2) > 1
Therefore, the correct answer is tan(x/2) = 3
