1/L1 = cos(x)
8/L2 = sin(x)

=> L1 + L2 = 1/cos(x) + 8/sin(x)
d(L1 + L2)/dx = sin(x)/(cos(x)^2) - 8cos(x)/(sin(x)^2)

let d(L1 + L2)/dx = 0
=> sin(x)/(cos(x)^2) = 8cos(x)/(sin(x)^2)
=> sin(x)^3 = 8*cos(x)^3
=> sin(x) = 2cos(x)
=> tan(x) = 2
=> L1 = sqrt(1^2 + 2^2) = sqrt(5)
=> L2 = sqrt(4^2 + 8^2) = sqrt(80)

Proofs of Derivative of Trig Functions