Answer: To prove the identity cosh 3x = 4 cosh^3 x - 3 cosh x, we'll use the definition of hyperbolic cosine (cosh) and the identity cosh^2 x - sinh^2 x = 1.
Definition of hyperbolic cosine:
cosh x = (e^x + e^(-x)) / 2
Step 1: Start with the left-hand side (LHS) of the identity:
LHS = cosh 3x = (e^(3x) + e^(-3x)) / 2
Step 2: Now, we'll express cosh 3x in terms of cosh x using the identity cosh (3x) = cosh^3 (x) + 3cosh(x)sinh^2(x):
cosh 3x = cosh^3 x + 3cosh x sinh^2 x
Step 3: Next, we'll use the identity cosh^2 x - sinh^2 x = 1 to replace sinh^2 x with cosh^2 x - 1:
cosh 3x = cosh^3 x + 3cosh x (cosh^2 x - 1)
Step 4: Now, we'll factor out the common term cosh x from the last two terms:
cosh 3x = cosh^3 x + 3cosh^3 x - 3cosh x
Step 5: Combine like terms:
cosh 3x = 4cosh^3 x - 3cosh x
Thus, we have proven the identity cosh 3x = 4cosh^3 x - 3cosh x.