Prove the following identity,

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

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Stumbling Robot

A Fraction of a Dot
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Tag: Identities

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Prove that *coth*^{2} x – csch^{2} x = 1

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Prove that * tanh*^{2} x + sech^{2}x = 1

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Prove that *2 cosh*^{2} (x/2) = cosh x + 1

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Prove that *2 sinh*^{2} (x/2) = cosh x – 1

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Prove that *(cosh x + sinh x)*^{n} = cosh (nx) + sinh (nx)

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Prove that *cosh x – sinh x = e*^{-x}

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Prove that *cosh x + sinh x = e*^{x}

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Prove that *cosh (2x) = cosh*^{2}x + sinh^{2} x

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

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Prove a formula for *cosh (x+y)*

Prove the following identity,

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

Prove the following identity,

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

Prove the following identity,

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

Prove the following identity:

* Proof. * We compute directly from the definition of ,

Prove the following identity:

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

Therefore, we have

Prove that

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

Prove that

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

Prove that

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