# Mixing Hyperbolic Functions and Logarithmic Functions, and using them to prove equalities using the definition of Hyperbolic Functions

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?

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

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