How do i solve this interesting arctangent product problem?

Question

X² - sqrt(8)*x + 1 = 0

Calculate (the exact value, without using a calculator):

arctan(x1)*arctan(x2)

Where x1 and x2 are the solutions of the inital ecuation.

The first requirement was: calculate arctan(x1)+arctan(x2)

Which was pretty easy, solved it by using tangent formula "tan(a+b)".

I guess the solution is similar but i don`t know where to start.

Answer 1

Solving x² - √8x + 1 = 0 with the quadratic formula results in x₁ = √2 + 1 and x₂ = √2 - 1

So we want arctan(√2 + 1)•arctan(√2 - 1)

I didn't have a clue what the exact value of either was so I've got to admit I cheated and just plugged tan-1(√2 + 1) and got 3π/8 and tan-1(√2 + 1) was π/8.

After I knew that, I first used the tangent half angle equation to find tan(π/8) by doing tan(π/4/2) = (1-cos(π/4))/sin(π/4) = √2 - 1

Then I used the tangent angle addition equation tan(3π/8) = tan(π/4 + π/8) = [tan(π/4) + tan(π/8)]/[1 - tan(π/4)•tan(π/8) = √2 + 1

So, arctan(√2 + 1)•arctan(√2 - 1) = 3π/8•π/8 = 3π²/64