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# Prove that coth2 x – csch2 x = 1

Prove the following identity,

Proof. From the definitions of hyperbolic cotangent and hyperbolic cosecant we have,

# Prove that tanh2 x + sech2x = 1

Prove the following identity,

From the definitions of hyperbolic tangent and hyperbolic secant we have,

# Prove that 2 cosh2 (x/2) = cosh x + 1

Prove the following identity,

Proof. Computing directly from the definition of the hyperbolic cosine,

# Prove that 2 sinh2 (x/2) = cosh x – 1

Prove the following identity:

Proof. We compute directly from the definition of ,

# Prove that (cosh x + sinh x)n = cosh (nx) + sinh (nx)

Prove the following identity:

Proof. We know from a previous exercise (Section 6.19, Exercise #9) that

Therefore, we have

# Prove that cosh x – sinh x = e-x

Prove that

Proof. Using the definition of the hyperbolic functions in terms of the exponential, we have

# Prove that cosh x + sinh x = ex

Prove that

Proof. We use the definitions of hyperoblic sine and cosine to compute,

# Prove that cosh (2x) = cosh2x + sinh2 x

Prove that

Proof. We know from this previous exercise (Section 6.19, Exercise #6) that

Therefore,

# Prove that sinh(2x) = 2 sinh x cosh x

Prove that

Proof. We know from this exercise (Section 6.19, Exercise #5) that

Therefore,

# Prove a formula for cosh (x+y)

Prove that

Proof. Using the definition of the hyperbolic cosine in terms of the exponential we have,