sin(13π/8) , is in quadrant IV so the angle will be negative, find the reference angle.
sin(13π/8) = -sin(16π/8 - 13π/8) = -sin(3π/8) = -cos(4π/8 - 3π/8) = - cos(π/8).
Use half angle formula for cos(x):
cos²(x/2) = (cos(x) + 1) / 2
Let x = π/4
cos²(π/8) = (cos(π/4) + 1) / 2
cos²(π/8) = (√(2) / 2 + 1) / 2
cos(π/8) = √(√(2) / 4 + 1/2)
-cos(π/8) = -√((√(2) + 2) / 4)