Got a question regarding using the definition of a hyperbolic function to prove other equalities based on a given equation.

Given: $$ x=\ln\left[\tan\left(\frac{\pi}{4}+\frac{\theta}{2}\right)\right] $$

Find: $$ e^x $$ and $$ \frac{1}{e^x} $$

And HENCE prove that: $$ \sinh x=\tan(\theta) $$

I know I need to use inverse hyperbolic functions to solve this, but I don't know where the definition comes in. Can someone maybe explain it to me?

  • $\begingroup$ What did you try? Can you compute $e^x$? Hint : $\forall a>0,\;e^{\ln a}=a$. $\endgroup$ – paf Aug 10 '16 at 10:17
  • $\begingroup$ I got it. Thanks $\endgroup$ – Rene Aug 10 '16 at 10:36

Since the definition of hyperbolic function: $$\sinh x = \frac{e^x-e^{-x}}{2}$$ , $$\sinh x = \frac{\tan(\frac{\pi}{4}+\frac{\theta}{2})-\frac{1}{\tan(\frac{\pi}{4}+\frac{\theta}{2})}}{2} = \frac{(\tan(\frac{\pi}{4}+\frac{\theta}{2}))^2-1}{2{\tan(\frac{\pi}{4}+\frac{\theta}{2})}} = -1/\tan (\pi/2 + \theta) = \tan(\theta) $$

  • $\begingroup$ Awesome. Thanks. Seems so simple. $\endgroup$ – Rene Aug 10 '16 at 10:37

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