source  observer
t_{0} = 0 yr  0 yr +  \frac{100 lyr}{c}

t_{1} = 1 yr  1 yr + \frac{100 lyr − (1 yr)v cos(θ)}{c}

t_{2} = 2 yr  2 yr + \frac{100 lyr − (2 yr)v cos(θ)}{c}

interval =  t_{1}  −  t_{0}

= [1 yr + \frac{100 lyr − (1 yr)v cos(θ)}{c}] − \frac{100 lyr}{c}

= (1 yr) [1 − \frac{v cos(θ)}{c}]

distance = v(1 yr) sin(θ)

apparent speed = \frac{distance}{interval}

= \frac{v(1 yr) sin(θ)}{(1 yr) [1 − (v/c) cos(θ)]}

= \frac{v sin(θ)}{[1 − (v/c) cos(θ)]}

apparent speed = \frac{v sin(θ)}{[1 − (v/c) cos(θ)]}