We have a^x + a^(-x) = 2.cosh(x), so here cosh(a) = 1000/2 = 500. As cosh(2.a) = 2.((cosh(a))^2)-1, we have cosh(2.a) = 2.(500^2) -1 or cosh(2.a) = 499999, and finally a^(2.a) + a^(-2.a) = 2.cosh(2.a) = 2.499999 = 999998. Simpler: As (X^2) + ((1/X)^2) = ((X + (1/X))^2) -2, so here with X = a^1000 we have a^2000 + a^(-2000) = (((a^1000) + a^(-1000))^2 -2 = (1000^2) - 2 = 999998
We have a^x + a^(-x) = 2.cosh(x), so here cosh(a) = 1000/2 = 500. As cosh(2.a) = 2.((cosh(a))^2)-1, we have cosh(2.a) = 2.(500^2) -1
or cosh(2.a) = 499999, and finally a^(2.a) + a^(-2.a) = 2.cosh(2.a) = 2.499999 = 999998.
Simpler: As (X^2) + ((1/X)^2) = ((X + (1/X))^2) -2,
so here with X = a^1000 we have a^2000 + a^(-2000) = (((a^1000) + a^(-1000))^2 -2 = (1000^2) - 2 = 999998
elevando al quadrato la prima espressione e il risultato si ottiene: 1000^2 - 2 cioè 999998
a¹⁰⁰⁰ + a-¹⁰⁰⁰ = 1000
a²⁰⁰⁰ + a-²⁰⁰⁰ = ?
a¹⁰⁰⁰ + a-¹⁰⁰⁰ = 1000
a¹⁰⁰⁰ + 1/a¹⁰⁰⁰ = 1000
Let, a²⁰⁰⁰ + a-²⁰⁰⁰ = k
k = a²⁰⁰⁰ + a-²⁰⁰⁰
= (a¹⁰⁰⁰)² + (1/a¹⁰⁰⁰)²
= (a¹⁰⁰⁰ + 1/a¹⁰⁰⁰)² - 2
= (1,000)² - 2
= 1,000,000 - 2
= 999,998
elevando al quadrato la prima espressione e il risultato si ottiene: 1000^2 - 2 cioè 999998