How to Derive Double Argument Properties of Hyperbolic Functions

Derive the Double Argument Formula for Sinh(x)

• 1

  Take the definition of the hyperbolic sine, and replace all the x's with (2x) to get: sinh(2x) = -i sin(2ix).

• 2

  Replace a with ix in the double angle formula of the sine function to get: sin(2ix)=2sin(ix)cos(ix). Substitute the right-hand term in the original equation to get: Sinh(2x)=-i*sin(ix)cos(ix)=2*(-i)*sin(ix)cos(ix).

• 3

  Change the sines and cosines back into hyperbolic sines and cosines. Since cos(ix)=cosh(x) and -isin(ix)=sinh(x), you will arrive at the double argument formula: sinh(2x)=2sinh(x)cosh(x).

Derive the Double Argument Formula for Cosh(x)

• 4

  Double the arguments in the definition of the hyperbolic cosine: cosh(2x) =cos (2ix).

• 5

  Use the double-angle identity for cosines to switch out cos(2ix): cos(2a) = 1 – 2sin²(a). So: cos(2ix)=1 – 2sin²(ix). Replace that into the original equation to get: cosh(2x)=1-2sin²(ix).

• 6

  Switch out sin(ix) for-sinh(x)/i , using the definition of the hyperbolic sine: cosh(2x)=1-2*(-sinh(x)/i)².

• 7

  Multiply the i's in the denominators and simplify to get the double argument formula:
  cosh(2x)=1-2*(-sinh(x)/i)²=1+sinh²(x)

Derive the Double Argument Formula for the Hyperbolic Tangent

• 8

  Double the arguments in the definition of the hyperbolic tangent: tanh(2x)=sinh(2x)/cosh(2x).

• 9

  Swap in the double argument formulas for hyperbolic sine and cosine (which you just derived):
  tanh(2x)=2sinh(x)cosh(x)/(1+sinh²(x))

• 10

  Use the identity 1=cosh²(a)-sinh²(a) and substitute it into the denominator. Simplify to get: tanh(2x)= 2sinh(x)cosh(x)/(cosh²(x)+sinh²(x)).

• 11

  Divide the numerator and denominator each by cosh²(x). Simplify the equation with the definition of the hyperbolic tangent and you will get the double argument formula:
  tanh(2x)=2tanh(x)/(1+tanh²(x)).