exact value of tan(arccos((x−1)/8))

Draw a right triangle with sides labeled so that

cosθ = (x-1)/8
That is, the adjacent side is x-1 and the hypotenuse is 8
The opposite side is therefore √(8^2 - (x-1)^2)
That makes tanθ = (x-1)/√(8^2 - (x-1)^2)