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Prove cosh 3x = 4 cosh^3 x - 3 cosh x

User Jane S
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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.

User Lidia
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