OK I did a force body diagram, with T to the right, F to the left, N upwards and T + ΔT at an angle Δθ to the horizontal.
Using small angle approximation,
(T + ΔT)Δθ = N,
μ(T + ΔT)Δθ = F
(T + ΔT)cosΔθ = T - F, small angle approx,
T + ΔT = T - F
ΔT = F
μ(T + ΔT)Δθ = ΔT
If the μΔTΔθ part...