To solve this exercirse, you must apply the proccedure below:
1. You need to apply the "Half-angle identity for tangent", which is:
Tan(θ/2)=Sinθ/1+Cosθ
2. But first, the angle 5π/4 must be expressed as a product of 1/2, as below:
5π/8=(5π/4)(1/2)
3. Now, you can susbtitute the angle into the formula:
=Sinθ/1+Cosθ
=Sin(5π/4)/1+Cos(5π/4)
Sin(5π/4)=-√2/2
Cos(5π/4)=Cos(π/4)
π/4 is the reference angle of 5π/4
π/4=√2/2
4. Then:
=(-√2/2)/(1-√2/2)
5. When you simplify the expression, you obtain:
=-1-√2
6. Therefore, the answer is: -1-√2