Consider a 4-year amortizing loan. If i borrow $1,000 initially, and
repay it in four equal annual year-end payments and the interest rate
is 8 percent, how can i show that the annual payment is $301.92?

Below is the formula for figuring this out.

http://www.hughchou.org/calc/formula.html

"First you must define some variables to make it easier to set up: 
P = principal, the initial amount of the loan 
I = the annual interest rate (from 1 to 100 percent) 
L = length, the length (in years) of the loan, or at least the length
over which the loan is amortized.
The following assumes a typical conventional loan where the interest
is compounded monthly. First I will define two more variables to make
the calculations easier:
J = monthly interest in decimal form = I / (12 x 100) 
N = number of months over which loan is amortized = L x 12"

You can use this for yearly payments as well.  Just use the number of
years for "N" and the annual interest rate for "J".

M = P * ( J / (1 - (1 + J) ** -N))

M = 1000 * (.08 / (1 - (1 + .08) ** -4))

M = 1000 * (.08 / (1 - (1.08) ** -4))

M = 1000 * (.08 / (1 - .73503))

M = 1000 * (.08 / .26497)

M = 1000 * .30192

M = 301.92

If you need any further explanation please let me know by posting a
request for clarification.

Thanks!

Can you clarify, please?  What do you mean by "How can I show that the
annual payment is $301.92?"  Do you mean that you know that is
definitely the payment?  Or is that the payment you're hoping for?  Or
do you really want to know what the 4 equal payment amounts will be?