tan(x/8) is the root of f(x), an 8th degree monic polynomial with integer coefficients.

What is the value of f(1)?

It's not entirely clear to me what tan(x/8) means in this problem.

If tan(x) = 2, then geometrically, the corresponding triangle is 1,2, sqrt(5). tan (x/8) is then around 0.139283952438412.

There are some very tedious ways of working back from the root to the polynomial, but there is an implementation RootApproximant[] in WolframAlpha that does the heavy lifting.

However these figures do not render a monic polynomial in the 8th degree.

On the other hand, using y=x/8 and not tan (x/8) gives the slightly larger RootApproximant[0.1383935897242,8], producing the monic form x^8+15x^7-10x^6-10x^5+4x^4+24x^3-5x^2- 7x + 1, with f(1) worth 13.