When attempting to prove the formula for Compton Scattering, I obtained the following equation(I have omitted the details on how the equation is obtained as it is not relevant to this post):
$$f = f' + k(f^2 - 2ff'\cos\theta +f'^2)$$
where $~k~$ is a constant (I again omitted the physical meaning of the variable as it is not relevant).
The formula that I would like to obtain is the following: $$\lambda' - \lambda = \frac{k}{c}(2-2\cos\theta)$$ where $\lambda= \frac{c}{f}$.
If working backward, we can also obtain an expression in terms of $f$: $$cf - cf' = \frac{k}{c}ff'(2-2\cos\theta)$$
The answer key derives the formula for first converting $f$ to $\lambda$ and use the trick of $\Delta = \lambda'-\lambda$.
So here are my following questions:
$(1).~~$ Why does using the trick $\Delta = \lambda'-\lambda$ work to prove the formula? Looking back on the problem, is there something I would have been able to see that suggests that this is the way to go?
$(2).~~$ If I don't do the conversion(from $f$ to $\lambda $), I couldn't seem to get the formula that I wish to prove although these two quantity should be equivalent... why?
Below are my attempt to prove the formula in terms of $f$:
with the starting equation, divide everything by $f^2$:
$$\frac{1}{f} - \frac{f'}{f^2} = k(1-\frac{2f'}{f}\cos\theta + \frac{f'^2}{f^2})$$.
And now we replace $f'$ with $f+ \Delta$:
$$\frac{1}{f} - \frac{\Delta + f}{f^2} = k(1-\frac{2(f+\Delta)}{f}\cos\theta + \frac{(f+\Delta)^2}{f^2})$$
If we clear things up, we get:
$$-\frac{\Delta}{f^2} = k(2-\frac{2(f+\Delta)}{f}\cos\theta + \frac{2\Delta}{f} + \frac{\Delta^2}{f^2})$$
Here I am stuck - I couldn't seem to simplify this expression any further.