Very simple...

We can make it very simple in relativistic geometry.
 Let us try to determine the variation of a wave by change of frame of reference.
 Let us take the Poincaré transform relating to time.
 We have To'=(To-x.Vo/c²)/sqrt(1-Vo²/c²)
 Let To'=(To-[To/To]x.Vo/c²)/sqrt(1-Vo²/c²)
   We have To'= To (1-(x/To).Vo/c²)/sqrt(1-Vo²/c²)
   Let also To'= To (1-[x/c.To].(Vo/c))/sqrt(1-Vo²/c²)
   However, [x/c.To] = - cosα (negative To).
   Hence To'= To.(1+cosα.Vo/c).sqrt(1-Vo²/c²)
   Now, λ/To = λ'/To' = c
   Let λ'=λ.sqrt(1-Vo²/c²)/(1+cosα.Vo/c)
   This was to be demonstrated.
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 R.H.