tan 5pi/8 exact value

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