Question

prove that:
cosh3x=4cosh^3x-3coshx

Answer 1

Answer:

cosh 3x = (e^(3x) + e^(-3x) ) / 2

cosh 3x = (1/2) e^(3x) + (1/2) e^(-3x) ----------- (1)

4 cosh^3 x - 3 cosh x = 4 ( (e^(x) + e^(-x) ) / 2)^3 - 3 ( e^(x) + e^(-x) ) /2

= (4/8) ( e^x+ e^(-x))^3 - 3 ( e^(x) + e^(-x) )^2

(a+b)^3 = a^3 + 3a^2 b + 3 a b^2 + b^3

(a+b)^2 = a^2+ 2ab + b^2

4 cosh^3 x - 3 cosh x

= (1/2) ( e^(3x) + 3 e^(2x) e^(-x) + 3 e^(x) e^(-2x) + e^(-3x) ) - 3 ( (e^(x) + e^(-x) )/2)

= (1/2) e^(3x) + (3/2) e^(x) + (3/2) e^(-x) + (1/2) e^(-3x) -(3/2) e^(x) - (3/2) e^(-x)

= (1/2) e^(3x) + (1/2) e^(-3x) + (3/2) e^(x) + (3/2) e^(-x) - (3/2) e^(x) - (3/2) e^(-x)

= (1/2) e^(3x) + (1/2) e^(-3x) -------------- (2)

(1) = (2)