Q.419. Hyperbolic trigo functions - proof.

Question 419.
If tanh(x/2) = tan(x/2), prove that coshx cosx = 1.

Answer 419.
tanh (x/2) = tan(x/2)
=> tan(x/2)
= sinh x / cosh x
= [(1/2) {e^(x/2) - e^(-x/2)}] / [(1/2) {e^(x/2) + e^(-x/2)}]
= [e^(x/2) - e^(-x/2)] / [e^(x/2) + e^(-x/2)] ... ( 1 )

cosx
= (1 - tan^(x/2)) / (1 + tan^(x/2))
= [1 - {e^(x/2) - e^(-x/2)}^2/{e^(x/2) + e^(-x/2)}^2] / [1 + {e^(x/2) + e^(-x/2)}^2/{e^(x/2) + e^(-x/2)}^2]
.................[plugging the value of tan(x/2) from ( 1 )]
= 4 / 2 (e^x + e^-x)
= 2 / (e^x + e^-x)
= 1 / cosh x
=> cosh x cosx = 1.

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