In 1995, the age ratio between a father and son is the difference between sin theta and cos theta

In 1995, the age ratio between a father and son is the difference between sin theta and cos theta.theta is the angle between 5 and 71 in a snakes and ladders boardl.8 years later the ratio is the square root of the raion of their ages 4 years earlier.What are their ages?

i) It is confusing to take the angle θ; whether to take it as tan(θ) = 7/5 or 8/6; Anyway, it is taken as 8/6, since the solution of which only will give ratio of ages as rational numbers; other one will give ratio as irrational number.

ii) When tan(θ) = 8/6 = 4/3, sin(θ) = 4/5 and cos(θ) = 3/5
==> Difference of them = 1/5
So ratio in ages of father:son = 5:1
Hence, let at present the age Father be 5x years and that of his son be x year.

iii) After 8 years, their ages would be 5x+8 & x+8 years respectively.
Hence their ratio = (5x+8)/(x+8) ---------- (1)

iv) Four years earlier their ages were, 5x-4 & x-4 years respectively.
Hence their ratio = (5x-4)/(x-4) ------------ (2)

v) As given from (1) & (2), (5x+8)/(x+8) = √{(5x-4)/(x-4)}
Squaring, (5x+8)²/(x+8)² = (5x-4)/(x-4)

==> [(5x+8)² + (x+8)²]/[(5x+8)² - (x+8)²] = (5x - 4 + x - 4)/(5x - 4 - x + 4)
[Application of compenendo-dividendo property of ratios]

Cross multiplying and simplifying the above, it reduces to:
5x² - 24x - 128 = 0
Factorizing, (x - 8)(5x + 16) = 0
==> Either x = 8 or x = -16/5

But x = -16/5, will result in ages being negative. Hence it is discarded.
So x = 8

Thus the age of father is 40 years and that son is 8 years.