Originally Posted by fares A car with a mass of 1000 kg traveled in a straight line at a speed of v0 = 21m / sec.
Once t = 0 the driver was moved to neutral and continued to travel in a straight line.
From this moment until the car stopped its speed was measured and found suitable for the function
v (t) = 10tan (5⋅103t) m / sec 
What does the "" mean here? Is that just a negative? If so just one  would do. And is "5⋅10" "5 times 10"? If so why not just write "50"?
That is, is v(t)= 10 tan(50 3t)? Or do you mean v(t)= 10 tan(5(10 3t)) but forgot the parentheses?
(When time is measured in seconds and the argument of the tan function is in radians)
A. Find the value of the constant.

What constant are you referring to? There is no undetermined constant in your formula.
B. Find the equivalent force that acts on the car as dependent on time and show that it can be displayed as F = αv2f0 as the momentary velocity of the car.

Force equals mass times acceleration. Find the acceleration by differentiating the velocity function and multiply by the mass of the car, 1000 kg. And, again, if v2 is intended to mean "v squared", write "v^2".
Find the constant values α and f0 (can be used for identity 1 + tan2 = = 1cos2))

Surely you do not mean "tan(2)" and "cos(2)"! Use ^ to indicate exponents. And do not write functions without their arguments: 1+ tan^2(x). 1+ tan^2(x) is equal to sec^2(x) which apparently are writing as 1/cos^2(x) but forgot the "/"!
third. Find the total distance that the car will go until it stops (hint: use u = cosx and additional positions as needed to calculate the integral)

The total distance the car will go is $\displaystyle \int_0^T v(t)dt$ where T is the value you calculated in (A).
("A", "B", and "third"?)